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OSMANIA UNIVERSITY LIBRARY 

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MATHEMATICS OF MODERN ENGINEERING 



GENERAL ELECTRIC SERIES 

POWER SYSTEM STABILITY 

By Selden B. Crary. Volume 1: Steady State 
Stability. 291 pages. 152 figures. 5> by 8%. 
Cloth. 

FIELDS AND WAVES IN MODERN RADIO 

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MATERIALS AND PROCESSES 

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CIRCUIT ANALYSIS OF A-C POWER SYSTEMS 

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A SHORT COURSE IN TENSOR ANALYSIS FOR 
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MATHEMATICS OF MODERN ENGINEERING 

Volume I by Robert E. Doherty and Ernest G. 

Keller. 3 14 pages. 83 figures. 6 by 9. Cloth. 

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PUBLISHED BY JOHN WILEY & SONS, INC. 



MATHEMATICS OF 
MODERN ENGINEERING 

VOLUME I 



BY 

ROBERT E. DOHERTY 

M.A., MS., LL.D. 
President, Carnegie Institute, of Technology 

AND 

ERNEST G. KELLER 

M.S., PhD. (University of Chicago] 
Consulting Mathctnatiiian, Curt'm-Wri&hl Corporation 



One of a Series written in the interest of 

the General Electric Advanced Engineering 

Program 



NEW YORK 

JOHN WILEY & SONS, INC. 
LONDON: CHAPMAN & HALL, LIMITED 



COPYRIGHT, 1936 
BY THE GENERAL ELECTRIC COMPANY 



All Rights Reserved 
This book or any part thereof must not 
be reproduced in any form without 
the written permission of the publisher. 



SEVENTH PRINTING, OCTOBER, 1949 



PREFACE 

The purpose of this text is to facilitate the study and use of mathe- 
matics with especial reference to its applications in engineering. The 
justification of another text in this general field, however, is not its 
purpose merely, which is broadly the same as that of any book on the 
same subject, but the point of view from which it is written. The 
latter has been tempered, first, by the joint participation of a mathe- 
matician who has worked in engineering and of an engineer who has 
worked with mathematics; and secondly, by the atmosphere of the 
engineering office and the classroom. On its educational side, the text 
represents a gradual development of material and form during a class- 
room experience of ten years; and, on the engineering side, embodies a 
scope and method suggested by engineering work extending over a 
much longer period. It is thus neither a text on mathematics nor 
one on engineering, nor yet merely a handbook on engineering mathe- 
matics. Rather, it is, first, a guide in bridging the gap in engineering 
between physics and mathematics by the scientific method; and 
secondly, a presentation, suitable for engineers, of those aspects of 
mathematics which the experience of a large manufacturing organiza- 
tion in dealing with electrical and mechanical problems has indicated 
to be of value to engineers. 

Such a book may appear choppy and lacking in unity from the 
mathematician's point of view. However, the unity of the text lies 
in its method of approach in the mathematical formulation of engineer- 
ing problems. The problems themselves and the mathematical devices 
used in solving them afford a great variety, and it is this variety that 
may, in a superficial review, present the appearance of choppiness. 

We believe that there is a broad educational field for a text which 
outlines the application of mathematics from the point of view referred 
to, and which is of proved usefulness in class work. The text was 
originally developed for the Advanced Course in Engineering of the 
General Electric Company. Although prepared for that course, 
which comprises a selected group of engineering graduates just out of 
college, nevertheless the book should be useful as well in college in 
both undergraduate and graduate engineering work. This volume 
covers ground which those students who are interested in the higher 



VI PREFACE 

levels of engineering service should study as undergraduates. The 
material of the second volume fits more appropriately in graduate 
work. The only knowledge presupposed for understanding Volume I 
is the calculus. 

On account of the special nature and purpose of the text it has 
seemed desirable to include a Foreword for Instructors which may be 
helpful to them in presenting the point of view to the student. 

Assistance from engineers of the General Electric Company is 
acknowledged. They have suggested problem material from actual 
engineering work, and offered valuable criticisms. We are especially 
indebted to Dr. A. R. Stevenson and Messrs. P. L. Alger, E. E. 
Johnson, and Alan Howard of the General Electric Company. 

Finally, we wish to express our thanks to Professor B. R. Teare, Jr., 
formerly of Yale University, and of the General Electric Company, 
now at Carnegie Institute of Technology, not only for his valuable 
suggestions but also for his assistance in the selection and presentation 
of material. 

New Haven, Connecticut ROBERT E. DOHERTY 

Schenectady, New York ERNEST G. KELLER 



FOREWORD FOR INSTRUCTORS 

The great increase in the use of mathematics in engineering during 
the last decade or so is merely one phase of a general increase through- 
out the whole field of applied mathematics. In the latter field, tensor 
analysis is now the everyday language of the physicist; Hamilton's 
principle, involving the calculus of variations, is one of the chief stand- 
by theorems of mathematical physics. Courant and Hilbert's Me- 
Ihoden der mathematischen Physik is quite as much a textbook of mathe- 
matics as it is of physics. The development of wave mechanics has 
made the concepts of groups, Hermitian matrices, Eigenwerte, ex- 
tremals, invariants, and contact transformations, and many other 
similar concepts, common property of the chemist as well as of the 
physicist and mathematician. Modern theories of ballistics are con- 
cerned almost entirely with the solution of certain differential equa- 
tions. Extensions have not stopped at the boundaries of physical 
science. The mathematical developments in political economy, statis- 
tics, and life insurance form a long list in the Encyklopadie der mathe- 
matischen Wissenschaften, and the applications in biology, and psy- 
chology become increasingly more numerous. 

The engineering phase of this growth is far-reaching. The advanced 
theories of periodic orbits have a counterpart in synchronous machine 
theory. The analogue of eccentricity of orbit is internal resistance, 
while the periodic coefficients of the differential equations introduced 
by rotating axes in special cases of the three-body problem are intro- 
duced in machine theory by the armature rotation. The non-linear 
equation of damped pendulum motion has an analogue in the study of 
synchronous machine stability, the hunting of the armature corre- 
sponding to pendulum motion. The quasi-differential and integral 
equations of Cotton are of service in the investigation of locomotive 
oscillations. Partial differential equations of the eighth order with 
boundary conditions expressed by differential equations of the fourth 
order make their appearance in connection with transformer oscilla- 
tions. Dyadics and tensors have been found of service in machine 
analysis and stability problems. Lagrange's equations are of impor- 
tance in mechanical engineering. Nearly every partial differential 
equation of not more than four independent variables is of some assist- 

vii 



iaii FOREWORD FOR INSTRUCTORS 

ance to the engineer. The calculus of variations is of use in many 
maximum problems. Vector analysis is of use wherever vector fields 
are found, especially in the discussion of vector magnetic potential 
within current-carrying regions and in the applications of Maxwell's 
equations. The Heaviside operational calculus is of almost as great 
importance in the heat-flow problems of the mechanical engineer as in 
the circuit problems of the electrical engineer. Statistical methods 
have been of service in sampling and in the investigation of windage 
loss in machines. Dimensional analysis has yielded much information 
where differential equation methods are too complicated. The theo- 
rems of Poincare on analytic differential equations and recent work on 
non-linear integral equations facilitate the struggle with non-linear 
problems of various kinds. The theory of functions of a complex vari- 
able is of extraordinary use in solving partial differential equations by 
means of conjugate functions. The theory of functions also puts the 
operational calculus on a rigorous basis in the simplest way. Practically 
all the well-known functions of applied mathematics are employed in 
the solution of problems in elasticity. The foregoing cases are perhaps 
sufficient to indicate the new importance which mathematics has come 
to have in engineering work, and they suggest as well the scope of 
modern engineering mathematics. 

The reason for this new importance is clear. Growing complexity 
in engineering problems has demanded facilities for their solution. 
Competition in industry tends to render all qualitative applied science, 
quantitative; complexity arises from the necessity of including more 
and more factors in the solution. There was a time, for instance, 
within the present generation when only two defined reactances were 
used in connection with applied theory of synchronous machines. 
There are now at least ten in common use, and still others in the theory. 
Another illustration of increased complexity is the problem of the riding 
characteristics of an electric locomotive. These characteristics depend 
simultaneously upon at least twenty-three parameters. The relative 
importance of each of these factors and its individual effect upon loco- 
motive motion can be determined only by analytical methods. As a 
final illustration, look back over the long span of the development of 
applied science in the field of electricity. In Bacon's time, the science 
of electricity consisted of an amusing collection of seemingly unrelated 
facts. Today most of the fundamental well-known relations of this 
science whether they be regarding the propagation of electromagnetic 
waves; calculation of space charge; generation, transmission, and use of 
power; determination of flux; calculation of eddy currents; the solution 
of potential ; the finding of reactance ; or the analysis of circuits are 



FOREWORD FOR INSTRUCTORS ix 

expressed by mathematical equations. Thus the commanding impor- 
tance of mathematical methods in modern engineering rests on the fact 
that they constitute indispensable facilities for dealing with the new 
complexities in practical problems. 

Mathematics facilitates the development of engineering science in 
a number of ways. In the first place, it does for engineering precisely 
what it does for each of the other fields mentioned above; it provides 
better methods of analysis; it is an essential adjunct to the scientific 
method. The latter is as much the method of the modern engineer as 
it is of the scientist, and it is important enough to warrant our digres- 
sing for a moment to discuss it. 

As conceived by Francis Bacon some three centuries ago, it con- 
sists essentially of four steps : 

1. Eliminate prejudices; or, in Bacon's words, "At the entrance 
of every inquiry our first duty is to eradicate any idol by which 
the judgment may be warped." 

2. Collect and study the data regarding the given situation. 

3. Project an hypothesis which, it seems, might rationalize the 
particular data. 

4. Test the hypothesis by applying it to the data, or to new 
data, thus comparing the calculated and actual values of quantities 
involved. Bacon says, "... but from the light of axioms, which 
having been educed from those particulars by a certain method 
and rule, shall in their turn point out the way again to new par- 
ticulars. . . . Our road does not lie on a level, but ascends and 
descends; first ascending to axioms, then descending to work." 

But it may be asked: What has mathematics to do with the scien- 
tific method? Bacon gave also the best answer to this question. 
"For many parts of nature can neither be invented with sufficient subtility, 
nor demonstrated with sufficient perspicuity, nor accommodated unto use 
with sufficient dexterity without the aid and intervening of mathematics'' 

A distinction should be drawn between the immediate objectives of 
the scientist and the engineer in their use of the scientific method. The 
one seeks primarily to establish new fundamental laws of science; the 
other, to predetermine the consequences of established laws in given 
situations. Yet in either case the approach is the same, and mathe- 
matics enters in the same way: induction leads to a generalization; 
deduction, in turn, leads to predicted particulars. Mathematics enters 
primarily in the second half of this " ascending " and " descending " 
process. 

A good illustration on the science side is the classic work of Kepler 
and Newton. Kepler studied Tycho Brahe's data regarding the mo- 
tion of planets, and was successful in framing an hypothesis that ration- 



X FOREWORD FOR INSTRUCTORS 

alized the data. The hypothesis was that during the elliptical motion 
of a planet the radius vector joining the focus of the ellipse to the planet 
traversed equal areas in equal intervals of time. This was half of the 
story. Newton completed it, at least for the time being that is, until 
Einstein. Newton carried the generalization further to inquire why 
planets should thus behave ; and after a study of all data at hand pro- 
jected the generalization that, if heavenly bodies attracted each other 
in direct proportion to the product of their masses and inversely as 
the square of their distances apart, a planet would have such a motion 
as Kepler had found it actually to be. Newton tested and proved the 
validity of his hypothesis by calculating from it the orbits of the moon 
and the planets and then comparing these with the observed orbits. 
Both Kepler and Newton followed the scientific method; they studied 
the situation, projected a generalization, " descended " mathemati- 
cally to the particular implications of that generalization, and then 
compared the calculated results with the actual. 

The engineering theorist follows precisely these steps, even if his 
objective is somewhat different. He collects and studies the data 
relating to the problem in hand; tentatively settles upon the funda- 
mental law or theory, already established by the scientists, from which, 
it seems to him, he will be able mathematically to deduce the desired 
particular relationships between the factors of his problem; sets up 
the equations accordingly, solves them if he can, and calculates mag- 
nitudes; and then he makes tests to determine whether his equations 
give reliable results and thus also whether they can be depended upon 
to predict, for instance, the performance of structures or apparatus of 
new design. 

Steinmetz's symbolic method of solving alternating-current circuits 
affords a good illustration. With the introduction of alternating cur- 
rents came the then difficult problem of calculating the performance of 
such circuits. After a study of available electrical data and a review 
of existing theory which might be useful, he projected the generaliza- 
tion that alternating currents and voltages, being approximately sine- 
wave phenomena, could be represented by the complex numbers. 
Applying this generalization, he set up equations representing currents 
and voltages, calculated their magnitudes and compared these with the 
actual ones, thus testing and proving the validity of the generalization. 

Having reviewed the significance of the scientific method and the 
relation of mathematics to it, we may now return to the consideration 
of other respects in which mathematics facilitates the development of 
engineering science. 

Demonstration or clearness of perception is, in certain instances, 



FOREWORD FOR INSTRUCTORS xi 

obtained by analogy, and a number of these in science are set forth by 
mathematics. For example, Laplace's equation is satisfied by the fol- 
lowing functions: .gravitational potential in regions unoccupied by 
attracting matter, electrostatic potential at points where no charge is 
present, magnetic potential in regions free from magnetic charges and 
currents, temperature of an isotropic medium in the steady state, 
velocity potential at points of a homogeneous non- viscous fluid moving 
irrotationally, and the real and imaginary parts of an analytic function. 
Moreover, the behavior of certain electric circuits, on the one hand, 
and of vibrating mechanical systems, on the other, is represented by 
the same differential equation. 

Mathematics has aided scientific discovery. Recall Maxwell's ob- 
servation that Ampere's law was inconsistent with the equation of 
continuity. Maxwell's change in the law led to the electromagnetic 
theory of light. Thus the validity or non- validity of hypotheses may, 
in certain cases, be demonstrated by mathematics. One of Heaviside's 
comments in this connection is interesting. " Faraday," he said, 
" . . . that great genius had all sorts of original notions wrong as well 
as right and, not being a mathematician, could not effectively dis- 
criminate." And again, Maxwell proved mathematically in 1864 the 
existence of electromagnetic waves; they were discovered by Hertz in 
1887. The " distortionless circuit " was treated by Heaviside in 1892, 
and the " loaded line," now used extensively in telephone circuits, 
was devised by Pupin in 1900. In fact, every solution of even an 
ordinary differential equation is a discovery in the limited sense that 
it reveals quantitatively how the various quantities are related to 
each other, even though it should describe no new phenomena. 

And, finally, mathematics affords a means of expressing scientific 
results that is at once compact and accurate. The equation or for- 
mula, indicating the numerical relationship between physical quantities 
in the problems of the engineer, is perhaps the phase of mathematics 
with which he is most familiar. It affords the means by which the 
engineering theorist can render easily and accurately available to the 
engineering practitioner the usable results of engineering science. With- 
out mathematics it would be practically impossible to describe ac- 
curately, for instance, the complex relationship between the voltage 
and current in a long transmission line, between the impressed forces 
and the magnitude of vibrations in resilient mechanical systems, or 
between loads and stresses in a beam. 

Thus, mathematics facilitates the development and application of 
engineering science by its being an essential adjunct to the scientific 
method; in clarifying thinking, correlating and interpreting data, 



xii FOREWORD FOR INSTRUCTORS 

aiding discovery, setting forth analogies, and expressing scientific 
results in usable form. 

We come now to the educational problem. In the authors' opinion 
there is one important phase of engineering education which is as yet 
inadequately developed, and its reasonable development will require a 
better understanding and a much greater use of mathematics than is 
now prevalent in the junior and senior years of engineering courses. It 
is the cultivation of the ability to analyze situations in terms of general 
principles. Mathematics is, as we have seen, an indispensable facility 
in such analysis and is thus correspondingly important in the educa- 
tional process. However, its use in engineering courses is now com- 
monly limited to only two things: utilizing the formula as a means of 
indicating how certain numerical calculations are to be made ; and call- 
ing upon the basic concepts of elementary mathematics trigonometric 
functions, rates, integrals, etc. in the study and understanding of the 
fundamental forms of engineering science. The use of mathematics as 
a tool in straight thinking is, in the authors* opinion, not at all what it 
should be. 

The reason for this seems clear. It is not, as is sometimes implied, 
that mathematics has been poorly taught; it is that engineering 
teachers do not make such use of it after the basic concepts and forms 
have been reasonably well taught. And the probable reason for this 
is the limitation of available time. The development of reasoning 
power requires time. Presumably it has seemed necessary to utilize 
practically all of the student's available time in his learning (memor- 
izing) the finished results of other minds and in developing certain 
routine engineering techniques; and thus little time has been left in 
which his mind could be guided into some independent thinking. 

Experience indicates the desirability of a different educational ap- 
proach, especially for the best minds. In practice, problems are set 
by situations, machines, and nature, and are not found definitely 
stated in words or equations. Hence, merely to remember some for- 
mulas and the type of problem to which they are applicable is of little 
avail in solving a new, practical problem. For the latter, not only is a 
knowledge of the basic engineering sciences necessary, but also a mind 
disciplined in sound reasoning. In this connection, it is significant 
that in the General Electric organization fifteen years ago practically 
all the engineering problems requiring real scientific analysis were 
referred to a very few individuals, most of whom had received their 
college training abroad. 1 Today in that organization there are a large 

1 F. C. Pratt, "Professional Engineering Education for the Industries," Journal 
of Engineering Education, Vol. 12, No. 5, January, 1922, pp. 227-235. 



FOREWORD FOR INSTRUCTORS xiii 

number of young men who can solve such problems. The difference is 
largely that these men, unlike their predecessors at the same age, have 
been disciplined in sound thinking in the Advanced Course in Engineer- 
ing of that organization. Moreover, they are exercising leadership 
not only in the highly technical sides of engineering but also in com- 
mercial engineering and in executive capacity. The thing that seems 
to count professionally is the cultivated intellect. The prominent 
place of mathematics and physics in the Advanced Course affords not 
merely a preparation for future specialized theoretical work involving 
these, but as well a rigorous discipline in analysis in the art of 
thought. It thus seems clear that such a discipline should begin more 
definitely in college than it does at present. 

How can this be accomplished, considering the present crowded 
undergraduate curriculum? Graduate study, which occurs as an 
answer on first thought, is not a complete solution for two reasons. 
In the first place, the student might not take graduate work; and 
secondly, even if he did, it would still be highly desirable to start a 
more definitely planned development of scientific thinking in under- 
graduate years in order for him to be properly prepared for graduate 
study on a genuinely scientific level. Only half of the educational job 
is done when subject-matter has been mastered; the other half is 
disciplining the mind in applying the acquired knowledge. To accom- 
plish this, the undergraduate curriculum for the better students should 
involve a gradual shift of emphasis during the junior and senior years, 
placing more and more upon scientific thinking and correspondingly 
less, in point of time, upon further extensions of the student's accumu- 
lation of memorized subject-matter. If the engineering graduate can 
think constructively and independently, he will readily acquire in pro- 
fessional practice the additional knowledge which, from time to time, 
he needs. However, if he has not developed scientific habits of thought, 
he will be embarrassed when he faces a real problem, notwithstanding 
an abundant knowledge of facts and forms. So the plan is to develop 
an undergraduate course, which will have a logical extension into 
graduate study, for the specific purpose of building up scientific habits 
of thought, exercise being obtained in the application of fundamental 
principles of mathematics, physics, and mechanics to the solution of 
physical problems. The course would begin where instruction in these 
subjects commonly ends at the beginning of the junior year and 
extend through the senior year, and also into graduate study, if taken. 

The present text is intended to fit into such an educational program, 
and thus certain observations regarding it should be noted. The first 
relates to its plan. Such a book must show the engineering student 



XIV FOREWORD FOR INSTRUCTORS 

how to use mathematics how to reduce the phenomena under observa- 
tion to mathematical equations, and then how to solve them. The 
first part of this process is the larger and usually the more difficult; 
when an engineering problem is reduced to equations, a long step has 
been taken toward the solution. Although naturally one cannot lay 
down detailed rules which would make possible the reduction of every 
engineering problem to equation form, nevertheless one can specify 
and emphasize the general steps in procedure according to which many 
problems can be so reduced, and thus aid the development of an orderly 
habit of thought. The procedure involves largely the subject-matter 
of physics. The method is outlined in Chapter I. 

The second part of the process involves principally the subject- 
matter of mathematics. The text should explain in a careful manner 
the mathematical theories and form of most use in engineering. More- 
over, it should contain a certain amount of information about mathe- 
matics. An engineer likes to know the underlying idea and purpose 
back of a mathematical concept or process before studying in exhaustive 
detail the process itself, just as a mathematician interested in learning 
induction-motor theory would appreciate a short general introduction 
describing the behavior of the machine before taking up the details of 
some special problem say the transient performance of the motor 
after a shock. In the second and subsequent chapters, an attempt 
has been made to present modern engineering mathematics from the 
above points of view. 

There are other observations which may render the character of 
the text more intelligible. The engineer's approach to a mathenuitical 
concept, function, or theory is different from that of the mathematician. 
It is not necessary for the engineer, as it is for the mathematician, to 
acquire an exhaustive knowledge of every mathematical function with 
which he deals. He needs to know a restricted definition and enough 
of its functional properties to use it intelligently. For illustration, he 
is not particularly interested in Bessel functions of fractional orders, 
although he is in those of positive integral orders. Moreover, the 
usefulness of the knowledge of the latter would not justify a semester's 
course in the subject. The engineer has complete confidence in the 
validity of established mathematical processes. Consequently proofs 
in themselves are of only secondary interest. Yet a book devoid of 
proofs is a handbook. Proofs do assist in the understanding and use 
of mathematics, and hence this text presents a limited number. It is 
frequently sufficient for the purpose in view to prove a special case of 
a theorem and then give the general statement. Because of its partial 
treatment of many functions, the text gives a limited number of 



FOREWORD FOR INSTRUCTORS XV 

references to fuller treatments of mathematical topics and to proofs 
omitted. 

Often the clearest approach to a mathematical concept, function, 
or theory is through the solution of an introductory problem, rather 
than through a formal approach by means of definitions, axioms, and 
theorems. Greenhill's Introduction to Elliptic Functions is a good illus- 
tration of this. An engineer is interested primarily in the application 
of mathematics to the solution of problems, and hence if he sees how 
some function or theorem is used in a problem he will be better able to 
understand the use. Thus the device of an introductory problem is 
utilized in the text. 

And finally, undergraduate classes using the text would presumably 
be led either by an instructor of mathematics or of engineering, depend- 
ing upon local conditions, although the more advanced work, including 
subject-matter which the student has not previously studied, would 
probably be given by an instructor of applied mathematics. 



CONTENTS 

PAGE 

PREFACE v 

FOREWORD TO INSTRUCTORS vii 

CHAPTER I 
MATHEMATICAL FORMULATION OF ENGINEERING PROBLEMS 

ARTICLE 

1. Deductive Method 2 

2. Coordinates 3 

3. Direction 3 

4. Units and Notation 5 

CHAPTER II 

BASIC ENGINEERING MATHEMATICS 
I. ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

5. Point of View 7 

6. Introductory Problem 8 

7. Origin of Some Differential Equations 10 

8. Definitions 10 

9. Principles of Mechanics 11 

10. Derivation of Differential Equations of Simple Mechanical Oscillations. . . 14 

11. Solution of Homogeneous Linear Differential Equations with Constant 

Coefficients 22 

12. First Method of Solution of Non-homogeneous Linear Differential Equations. 

Reduction by Differentiation to Homogeneous Form 26 

13. Initial Conditions. Damping. Harmonic Motion 31 

14. Summary 33 

15. Second Method of Solution of Non-homogeneous Linear Differential Equa- 

tions. Operator Method 34 

16. General Method of Solution of Non-homogeneous Linear Differential Equa- 38 

tions 

17. Summary 41 

18. Simultaneous Linear Differential Equations 44 

19. Electric Circuit Principles 47 

20. Derivation of Differential Equations of Simple Linear Circuits 49 

II. DETERMINANTS 

21. Introductory Problem 55 

22. Definitions 56 



xviii CONTENTS 

ARTICLE PAGE 

23. Laplace's Expansion 58 

24. Theorems Regarding the Expansion of Determinants 60 

25. Multiplication of Determinants 62 

26. Application of Determinants to Non-homogeneous Linear Equations 63 

27. Application of Determinants to Homogeneous Linear Equations 65 

28. Application of Determinants in Obtaining the Particular Integral or Steady- 

State Solution of Simultaneous Differential Equations with Constant 
Coefficients and Sinusoidal Applied Force 67 

29. Application of Determinants in Obtaining the Complementary Function in 

the Solution of Simultaneous Differential Equations with Constant Coeffi- 
cients 70 



III. FOURIER SERIES 

30. Definitions 73 

31. Introductory Problem 74 

32. Second Introductory Problem 75 

33. Values of the Fourier Coefficients a n , b n 76 

34. Fourier Series Expansion for the Interval to 2w 79 

35. Fourier Expansions in Sines or Cosines Only 79 

36. Fourier Series for the Interval / to / 81 

37. Harmonic Analysis 82 

38. Proof of Harmonic Analysis Rules 86 

39. Theory of Fourier Series 87 

40. Summary 91 

IV. SOLUTION OF HIGHER DEGREE AND TRANSCENDENTAL EQUATIONS 

41. Nature of Solutions of Algebraic Equations 95 

42. Newton's Method 95 

43. Successive Approximations 97 

44. Underlying Principle of Graeffe's Root-squaring method 98 

45. Preliminary Examples 99 

46. Graeffe's General Theory 105 

47. Rules for Graeffe's Method 125 

V. DIMENSIONAL ANALYSIS 

48. Uses and Nature of Dimensional Analysis 131 

49. Some Representative Results 133 

50. Checking Equations 138 

51. Change of Units 139 

52. Dimensional Constants 143 

53. Introductory Problem Leading to the IT Theorem 144 

54. The TT Theorem 145 

55. Principle of Similitude 155 

56. Systematic Experimentation 160 

57. An Additional Method 160 

58. Summary 161 



CONTENTS xix 

VI. GRAPHICAL AND NUMERICAL METHODS OF SOLVING 
DIFFERENTIAL EQUATIONS 

ARTICLE PAGE 

59. Nature of Numerical Integration 163 

60. The Differential Equation di/dt = }(i\ t) 164 

61. The System of Differential Equations dx/dt = f(x, y), dy/dt = g(x, y) 170 

62. The Radius of Curvature Method 173 

63. Preliminary Ideas for the General Method of Numerical Integration 176 

64. Reduction of Systems of Equations to the Normal Form 179 

65. General Method of Numerical Integration of Differential Equations 180 

66. Summary 185 

CHAPTER III 

VECTOR ANALYSIS 

I. OPERATORS AND LAWS OF VECTOR ANALYSIS 

67. Vectors 189 

68. Nature of Vector Analysis 190 

69. Algebra of Vectors 190 

70. Line and Surface Integrals Involving Vectors 193 

71. Vector Operators 196 

72. Derivatives of Vector Quantities 196 

73. Gradient 198 

74. Divergence 200 

75. Curl 203 

76. Operator Formulas ; 204 

II. DERIVATION OF THE PARTIAL DIFFERENTIAL EQUATIONS OF 
MATHEMATICAL PHYSICS OR VECTOR FIELDS 

77. Some Vector Fields 206 

78. Preliminary Theorems 207 

79. The Partial Differential Equations of Mathematical Physics 208 

80. Equation of Heat Conduction without Sources 209 

81. Equation of Heat Conduction with Sources 210 

82. Concept of Potential and Theorems of General Vector Fields 211 

83. Partial Differential Equations of Gravitational, Electrostatic, and Magneto- 

static Fields 213 

84. Maxwell's Equations 216 

85. Euler's Equation for the Motion of a Fluid 220 

86. Nature of the Solution of Partial Differential Equations 221 

87. The Partial Differential Equations of Electromagnetic Waves 222 

III. VECTOR FIELDS (VECTOR MAGNETIC THEORY) 

88. Experimental Basis of Magnetic Theory 224 

89. Force between Moving Charges 225 

90. Vector Magnetic Potential 226 

91. Integral Definition of Vector Potential 226 

92. Partial Differential Equation Definition of Vector Potential 227 



XX CONTENTS 

ARTICLE PAGE 

93. Vector Potential for Infinite Conductors 228 

94. Engineering Examples 231 

95. Additional Vector Relations in Vector Magnetic Theory 233 

96. Summary 234 

IV. DYADICS 

97. Definitions 235 

98. Digression from Definitions to a Physical Example 235 

99. Definitions Resumed 236 

100. Theorems 237 

101. Applications 238 

CHAPTER IV 
HEAVISIDE'S OPERATIONAL CALCULUS 

I. PRELIMINARY MATHEMATICAL PRINCIPLES LINE INTEGRALS IN THE 
COMPLEX PLANE 

102. Some Engineering Problems Solvable by Operational Calculus 240 

103. Historical Note 242 

104. Complex Numbers and Functions 242 

105. Definition of a Line or Curvilinear Integral 244 

106. Green's Formula 247 

107. Analytic Function of a Complex Variable 248 

108. Cauchy's First Integral Theorem and Its Corollaries 249 

109. Definitions of Certain Elementary Functions of a Complex Variable 250 

110. Integrals and Derivatives of Elementary Functions 252 

111. Taylor's Series and Singular Points 253 

112. Singular Points 254 

113. Laurent's Expansion 255 

114. Residues and the Residue Theorem 256 

II. RIGOROUS AND SYSTEMATIC PROOFS OF 
HEAVISIDE'S RESULTS 

115. Heaviside's Circuit Problems 261 

116. Bromwich's Solution of Heaviside's Unextended Problem 262 

117. Operational Formulas 266 

118. Heaviside's Expansion Theorem 267 

119. Fractional Powers of p 268 

120. Heaviside's Third Rule 272 

121. Heaviside's Extended Problem 272 

122. Summary 275 

III. MANIPULATIVE DEVICES AND APPLICATIONS OF THE 
OPERATIONAL CALCULUS 

123. Algebraic Operations and Shifting 277 

124. The Derivation of the Partial Differential Equations of the Transmission 

Line 280 



CONTENTS xxi 

ARTICLE PAGE 

125. Transmission Line Problems 282 

126. Partial Differential Equations of Linear Heat Flow 287 

127. Refrigerator Box Heat Leakage 288 

128. Water-wheel Generator Brake 291 

129. Switching 295 

REFERENCES 300 

ANSWERS 305 

INDEX 311 



MATHEMATICS OF MODERN ENGINEERING 



CHAPTER I 

MATHEMATICAL FORMULATION OF ENGINEERING 
PROBLEMS 

The type of engineering problem which lends itself most readily 
to mathematical methods of solution is that which involves primarily 
deductive reasoning. In such a problem the requirement is to deter- 
mine the implications of some physical law in a given situation. 
Although of course the solution always involves as well the inductive 
process of settling upon what law to employ, and the scheme of solu- 
tion, it is nevertheless primarily the deductive thought which is 
facilitated by mathematics. The difference between reasoning, on 
the one hand, step by step from cause to effect, and, on the other, by 
the use of mathematics is that in the latter case the conclusions from 
the premises are found largely by manipulation of mathematical 
symbols according to definite rules. Another difference is that, if the 
mathematical rules are not violated, one can be assured, as one cannot 
be in the non-mathematical procedure, that no factor included in the 
premises has been overlooked in the results; and thus that the results 
are as true as the premises. Moreover, as problems become more 
complex by the inclusion of more factors, a stage of complexity is 
reached beyond which a solution without the aid of mathematics is 
too difficult even for the best minds. 

However, all this is not to say that the non-mathematical pro- 
cedure is wholly useless or ineffective. On the contrary, it is absolutely 
essential in its place. It is complementary to the mathematical pro- 
cedure; without it, mathematics would be useless in engineering. 
In this field, at least, one cannot extract thought from mathematics 
without first having put some in. Moreover, the correctness of the 
solution cannot be tested and its engineering significance appraised 
without some thought apart from the mathematics. Thought and 
judgment there must be, and mathematics is an extremely effective 
tool to facilitate these. 



2 MATHEMATICAL FORMULATION OF ENGINEERING PROBLEMS 

Before mathematics can be thus used the problem must first be 
formulated in mathematical terms, and it is the purpose of the 
present chapter to outline the method for this. 

1. Deductive Method. There are certain essential, consecutive 
steps in the deductive method which lead to the mathematical forma- 
tion of a problem. The first is obviously to define the problem; the 
second, to settle upon a scheme of solution based upon some funda- 
mental physical law; the third, to state precisely the conditions which, 
according to that law, must be satisfied; and the fourth, to translate 
that statement into mathematical form. The entire procedure is 
illustrated in the formulation of the Introductory Problem, 6, 
Chap. II. 

Considered in more detail, the first step involves a definite descrip- 
tion, including a sketch, of the situation to be analyzed, and a state- 
ment of the result desired. 

The second step is one which, perhaps even more than the others, 
requiries a searching survey of one's resources in knowledge and 
imagination in order to settle upon the most likely scheme of attack. 
The scheme must include not only the fundamental law which is to 
serve as the basis of the solution, but also (1) the assumptions which 
are to be made in order reasonably to simplify the solution, and 
(2) a plan, however tentative and indefinite, according to which it 
appears that the desired results can be attained. The plan may, and 
usually does, involve the subsequent use of additional physical laws 
relating the quantities in hand to the independent variables. It is not 
always possible at the start to see clearly what relations will be required 
in carrying out a projected plan. To determine an effective scheme of 
attack is an exacting process, and often this object is not accomplished 
in the first trial. 

The third step demands clear and precise thought. It consists in 
reasoning out and stating in precise English the conditions which must 
be satisfied in accordance with the physical law settled upon as the 
basis of the solution. Such a statement constitutes the fundamental 
premise from which follow all the functional relationships in the solu- 
tion. Hence it is highly important that the statement should be 
most carefully and precisely formed. This statement is indeed the 
fundamental equation expressed in words. 

The final step in the formulation is the translation of that equation 
from words to mathematical symbols. All the different terms in the 
equation are physical quantities of the same kind the quantities may 
all be, for instance, forces or energies or momenta and for each term 
mu5t be substituted an equivalent mathematical term. Moreover, 



DIRECTION 3 

this equivalent must be in each case in terms of the chosen independent 
and dependent variables. For instance, in the simple problem of the 
motion of a body in a straight line, time is ordinarily taken as the 
independent variable, the displacement as the dependent variable. 
In order to determine such mathematical equivalents certain arbitrary 
conventions must be decided upon. 

2. Coordinates. In the first place an appropriate coordinate sys- 
tem must be chosen. This is a difficult step for most students. In- 
deed, it is not a simple matter, even for graduate students, in the more 
complicated problems, involving, for example, a system in which there 
may be two or more mutually dependent motions. The perplexing 
thing is not so much the concepts of the various forms of coordinates 
themselves as it is the problem of deciding upon what particular form 
to use and where to locate it in the given configuration representing 
the problem. A helpful principle is to select that form of coordinate 
system cartesian, spherical, cylindrical, etc. and to place it in that 
position which together will yield the simplest mathematical expres- 
sions. This process requires the same kind of visualization of possi- 
bilities and of their implications as does that of settling upon a general 
scheme of solution, referred to above. To carry out the process 
effectively in either case presupposes some experience. At first it 
may seem difficult not merely to choose the best coordinate system, 
but even to choose any at all which can be used to formulate the 
problem. However, a few thoughtful trials will provide some basis 
for judgment. To give some idea of the result of choosing different 
coordinates, example (a) of 10, Chap. II, is carried through for two 
different locations of the origin, a simpler differential equation being 
obtained in the second case than in the first. 

3. Direction. Another arbitrary choice must be made : what direc- 
tion will be considered as positive? In problems of motion, for in- 
stance, two choices must be made. One, which is essentially involved 
in the coordinate system, is the direction of positive displacement; 
the other, of positive force. Although the choice in either case is 
arbitrary, it should be made, like that of the origin of coordinates, 
so as to retain simplicity in relationships. Then after these choices 
have been made they must be adhered to rigorously. 

In problems of dynamics, forces are expressed as functions of dis- 
placement or of time or of both; and thus in order to make correct 
mathematical substitutions in the word-equation it is necessary to 
have not only the algebraic form of the term expressing the functional 
relationship in each instance, but also the proper algebraic sign of the 
term. The question of algebraic sign causes more needless worry in 



4 MATHEMATICAL FORMULATION OF ENGINEERING PROBLEMS 

the mathematical formulation of engineering problems than any other. 
Needless, simply because careless thinking, or no thinking, has been 
done in defining positive direction before the term is written down; 
then one wonders what the sign should be! However, with definite 
criteria established to which the question can be referred, one can 
proceed rationally. 

In simple problems of the motion of a body in a straight line it is 
customary, although not necessary, to consider force to be positive 
when it is in the direction of positive displacement. Under such an 
assumption, the definitions of the derivatives of displacement with 
respect to time, and the precise statement of the equation in words, 
together constitute the required criteria for settling the question of 
algebraic sign. Suppose, for illustration, that one of the forces acting 
on a body in motion along the #-axis is a " damping force," proportional 
to the velocity of motion. Such a force always opposes the motion, 
whatever may be the direction of the motion. The magnitude of the 
force is evidently 

dx 
k , 

dt 

dx 

where k is a coefficient of proportionality. The derivative is the 

dt 

time rate of change of displacement. When the motion is in the 

dx . 

direction of positive displacement is positive. Since we are express- 

dt 

ing a directed quantity, force, in terms of another directed quantity, 

dx 

velocity = , and these two quantities always have opposite senses, 
dt 

there must be a minus sign in front of the term if the positive direction 
of force is taken as the positive direction of displacement. The term 
thus becomes 



dt' 

It may be further emphasized that the minus sign does not mean that 
the force represented by the term is always directed toward negative 

dx 

displacement. Its sense is that only when has a positive sense. 1 In 

dt 

the case of a constant term, however, a negative sign before it does 
indicate, of course, that its sense is always opposite to the direction 

1 If the positive direction of force had been taken opposite to that of positive 
displacement, then obviously the algebraic sign of the term would have been plus. 



UNITS AND NOTATION 5 

defined as positive for the quantity. The question of the algebraic 
sign for damping and similar terms is fully illustrated in the Intro- 
ductory Problem, 6, Chap. II. 

In problems of the electric circuit, the directions of currents and 
voltages are relative to the physical elements of the circuit, and thus 
the respective positive directions are defined with reference to such 
elements. 

4. Units and Notation. In order for an equation to indicate cor- 
rect numerical relationships, either a standard set of units must be 
used; or, if any units are introduced in a problem that do not belong 
to the otherwise standard set employed, then a conversion factor must 
be used to compensate for the numerical difference they introduce. 
The classical example to illustrate this point is the relation between 
force /, mass -M, and acceleration a. This relation is expressed by the 
equation 

/ = kMa, 

where k is a constant depending upon the units assigned to the three 
quantities. For certain units, k is unity. Obviously, units can be 
arbitrarily assigned to any two of these three quantities, but if k is 
to be unity, the unit for the third is thus fixed. Certain standard sets 
or " systems " of units are in use in which k is unity: when / is in 
dynes, a in centimeters per second per second, and M in grams; 
when/ is in poundals, a in feet per second per second, and Mm pounds 
mass; when/ is in pounds force, a in feet per second per second, and 
M in slugs. 

Or we may look at the matter more fundamentally from the view- 
point of dimensions. It was learned in physics that all quantities 
relating to motion can be expressed dimensionally in terms of three 
properly chosen fundamental quantities; and the usual, though not 
necessary, choice of these is mass, length, and time. After the mag- 
nitudes of the three suitable fundamental units have been arbitrarily 
assigned, the units of other physical quantities are fixed so that con- 
stant factors are not required in most of the equations where the 
quantities are used. Such units comprise a consistent set, three 
illustrations of which are given above. 

Throughout this text the units relating to problems of motion are 
those based upon the fundamental units: 

mass in slugs, 
length in feet, 
time in seconds. 



6 MATHEMATICAL FORMULATION OF ENGINEERING PROBLEMS 

For illustration, in the physical law referred to above, a force of one 
pound will produce an acceleration of one foot per second in a mass of 
one slug. 2 The numerical value for mass is usually determined from 
the force which gravity exerts upon it, that is, its weight. Thus 



M 

M = , 
g 

where W is the weight in pounds and g the accleration in feet per 
second per second due to gravity. 

Closely related to the question of units is that of notation. Vague 
definition of mathematical symbols ranks next to the confusion of 
algebraic signs in causing the student, and also others, needless worry 
and errors in the mathematical formulation of a problem. There may 
be some connections in which a mathematical symbol can represent 
anything in general, but engineering is not one of them. Clean-cut 
and precise definition of each symbol, including the unit, is absolutely 
essential in the formulation, solution, and application of equations in 
engineering. 

2 A quantitative idea of this unit may be obtained from the fact that the mass of 
4 gallons of water is approximately one slug. 



CHAPTER II 
BASIC ENGINEERING MATHEMATICS 

The elementary mathematics of this chapter is basic to the solution 
of the more usual engineering problems, and is, moreover, fundamental 
to the study of the more advanced engineering mathematics of sub- 
sequent chapters. Most engineering graduates have studied some of 
the topics treated in the present chapter, but few have a working 
knowledge of all. 

I 
ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

Section I is concerned with the derivation and solution of ordinary 
differential equations with constant coefficients by the usual classical 
methods. Operational methods of handling systems of such equations, 
and the derivation and solution of the more complicated systems, are 
considered in later chapters. 

5. Point of View. We may distinguish between the engineering 
and the mathematical points of view. The study of ordinary differ- 
ential equations, from a mathematical point of view, consists of three 
parts : 

(a) A proof of the existence of a solution of a single equation or 
system of equations. 

(b) The investigation of the properties of the solution: continuity, 
differentiability, analyticity, and integrability of the solution with 
respect to both the independent variable and important parameters 
present. 

(c) The construction of a solution in a form suitable for the use 
at hand. 

The study of differential equations from an engineering point of 
view is different. It consists mainly of but two parts: 

(a) The derivation of the differential equation. 

(b) The solution of the differential equation. 

The existence of a solution is taken for granted from physical consider- 
ations and from the assumption that the differential equation repre- 
sents the physical problem under consideration. The engineering 

7 



8 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

point of view will be adhered to, especially in the treatment of this 
section. 

6. Introductory Problem. A simple problem may facilitate the 
understanding of differential equations. Following the approach out- 
lined in Chap. I, let us suppose that a body of mass M t constrained to 
move in a straight line, is acted upon by a spring as shown in Fig. 1. 
The spring may be both extended and compressed, and always acts 
on the body in the line of displacement. The motion takes place 
without friction. At the beginning of the period under consideration, 
when time / = 0, the body has such a position that the spring is 

undistorted, and is moving to the 
right with velocity v. It is desired 
to determine the equation of 
Position with motion of the body, that is, its 

spring undistorted^ . . , 1 

position at any time / subsequent 
to / = 0. 
^^^^ The solution may be based upon 

'//// 1 \'//////////////////. XT * > i a i r *. 

I *-H Newton s l second law ot motion. 

FIG. 1. According to that law the condition 

that must be satisfied is: the force 

applied to the body is equal to the product of its mass and acceleration, 
and the acceleration takes place in the direction of the applied force. 
This is the equation, stated in words, which later will be translated 
into mathematical terms. 

This requires the selection of a coordinate to designate the position 
of the body. Let x be the displacement of the body from such a posi- 
tion that the spring is undistorted, x being positive when the body is 

dx 

to the right of this position. The velocity is then ; the accelera- 

dt 

d' 2 x 

tion, . In accordance with the definition of the derivative as a 
Cit 

rate of change, is positive when the body moves to the right, 

d 2 x 

negative when it moves to the left; and is positive when the accel- 
eration is directed to the right, negative when it is directed to the left. 
Similarly, as discussed in Chap. I, the algebraic sign of the applied 
force denotes its direction. Let the convention for the sign of force 
be the same as for that of displacement, positive to the right, negative 
to the left. The spring exerts a force /, which is in the direction oppo- 

1 Newton's laws of motion are stated in 9. 



INTRODUCTORY PROBLEM 9 

site to the displacement, and, by Hooke's law, is proportional to the 
displacement. Thus 

/. = kx, 

where k is a coefficient which is constant if the elastic limit of the 
spring is not exceeded. 

Thus the previous equation stated in words becomes 

d 2 x 

M = - kx, 
dt 2 

or 

+*--* 

Eq. (1) is called the the differential equation of motion of the body. 
The solution, as determined by methods described in later sections, is 

\~k \~k 

x = A sin ^/ / + B cos -y/ /, (2) 

where A and B are arbitrary constants. That this is a solution may 
be verified by its substitution in Eq. (1). 

The arbitrary constants A and B are called constants of integra- 

dx 

tion. The conditions x = and = v at t = 0, given in the state- 

dt 

ment of the problem, are called initial conditions. By means of the 
initial conditions particular values of A and B can be determined so 
that (2) will give the position of the body at any time t subsequent to 
/ = 0. If (2) be differentiated with respect to time and then the 

dx 

conditions x = 0, = v, and t = be substituted in both (2) and the 
dt 

derivative of (2) there results 



(3) 



' = ^vi cos 

which may be solved for A and B, to obtain 



=v, 5=0. 

K 



10 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 
Substitution of these values in (2) gives 



Eq. (4) is the desired equation of motion, the answer to the problem 
stated. It is valid for the units given in 9. Reference to the above 
introductory problem will serve to illuminate some of the essential 
aspects of differential equations. 

7. Origin of Some Differential Equations. In the mathematical 
formulation of a problem, as discussed in Chap. I, the third step is a 
statement of the equation in words. If any of the quantities related in 
that statement involve rates of change, the mathematical equivalents 
are written in terms of derivatives and the resulting symbolic equation 
is a differential equation. In the example of the previous article one 
of the quantities related is the acceleration of the body, which in 

d 2 x 
symbolic form is . Thus Eq. (1) is a differential equation. 

dt 

Other physical quantities that involve rates and therefore lead to 
differential equations are numerous. For example, in geometric prob- 
lems slope and curvature depend upon derivatives. In electric-circuit 
theory, current is the rate of change of a charge, and induced voltage 
is proportional to the rate of change of magnetic flux linkages. 

8. Definitions. The following general definitions relate to differ- 
ential equations. An equation involving derivatives or differentials 
is a differential equation. An ordinary differential equation is one 
containing only ordinary or total derivatives as distinct from partial 
derivatives. The order of a differential equation is the order of the 
highest derivative which the equation contains. For example, 
Eq. (1) is of second order. The degree of a differential equation is the 
degree of the highest-ordered derivative which the equation contains 
after it has been cleared of radicals and fractions with respect to all 
derivatives. Eq. (1) is of first degree. A solution of an ordinary dif- 
ferential equation is a relation between the dependent and independent 
variables which satisfies the differential equation. A solution is called 
the general solution if it contains a number of arbitrary constants equal 
to the order of the equation, provided the constants enter into the 
solution in such a way that they cannot be replaced by a smaller 
number of equivalent arbitrary constants. Eq. (2) is the general solu- 
tion of Eq. (1). A particular solution of a differential equation is a 
solution obtained from the general solution by giving particular values 
to one or more of the arbitrary constants. Eq. (4) is a particular 



PRINCIPLES OF MECHANICS 11 

solution of Eq. (1). If the differential equation is of order n and the 
value of the dependent variable and the values of all derivatives up to 
and including those of order n 1 are known for a particular value 
of the independent variable, then the n arbitrary constants of the 
general solution can be determined. These values of the dependent 
variable and of the derivatives up to and including those of order n 1 
are called initial conditions or boundary conditions. In Eq. (1) 
n = 2, and the two Eqs. (3), which are statements of the initial condi- 
tions, were used to determine the constants. 

9. Principles of Mechanics. The physical principles with which 
we are first concerned fall into two groups; those pertaining to me- 
chanics and those relating to elementary electrical phenomena. 
Since mechanical forces and motions are, in general, more easily vis- 
ualized than electrical ones, we begin with the derivation and solution 
of the differential equations of some simple mechanical systems. 
Following this we shall consider the formulation of the differential 
equations of simple electric circuits. Since these equations are of the 
same type as the ones for mechanical systems, the solutions are of the 
same form and need not be discussed again. Such mechanical princi- 
ples as are used in Sec. I are briefly treated in this article. 

The whole science of kinetics rests upon Newton's laws of motion. 
These experimental laws may be stated as follows: 2 

1. A particle under the action of no forces remains at rest or moves 
in a straight line with constant speed. 

2. The resultant force acting on a particle equals in magnitude and 
direction the product of its mass by its acceleration. 

3. If two particles exert forces on each other, the force exerted by 
the first on the second (action) is equal and opposite to the force 
exerted by the second on the first (reaction). 

The first law is but a special case of the second. These two laws 
hold only with respect to a non-accelerated frame of reference, which, 
in most engineering problems, may be a frame that is fixed, or moving 
with a constant velocity, relative to the earth. 

The third law states that forces always occur in pairs and that the 
forces of a pair are equal in magnitude but act in opposite directions. 

The laws of motion for a particle may be extended to the case of a 
rigid body, which is nothing more than a group of particles. It is 
found that the resultant vector force is equal the product of the mass 
by the vector acceleration of the center of mass of the body. It is 
convenient, however, to employ a different statement of this law. 

* Page, Introduction to Theoretical Physics, courtesy of D. Van Nostrand Co., Inc. 



12 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

Suppose that a number of forces, /i, /2, . . ., act upon a body free to 
move in translation only. These forces may all be added vectorially 
to form a resultant which is the equivalent of the applied forces. 
Symbolically 



where 2/ represents the vector sum of /i, /2, . . . . If M is the mass 
of the body, and a is the acceleration (which has the direction of the 
resultant force), then by the second law 

/= Ma, 
or 

2/i = Ma. 
Thus 

- Ma + S/< - ~ Ma +/i +/ 2 + . . . = 0. (5) 



Now if the body were in static equilibrium the condition 

i = (6) 



would hold. This condition may be generalized to hold for any state 
of rest or translation by regarding Ma in (5) as one of the forces / 
in (6) acting on the body. Treating the equivalent force Ma, called 
the inertial reaction, as one of the applied forces in (6) is in accordance 
with D'Alembert's principle. 3 These equations hold for the forces 
and acceleration considered as vectors, and therefore also for the 
algebraic components in any direction. 

Besides these laws relating applied forces to inertial reaction, it is 
frequently necessary to employ laws relating certain of the forces 
other than inertial reaction to displacement and its derivatives. For 
example, in the introductory problem of 6 the relation 

/. = -** 

between the resilient force f a due to the spring, and the displacement 
x, was used. This is a statement of Hooke's law for elastic distortion. 
Another of frequent occurrence in mechanical problems is the approxi- 
mate relation between viscous damping force f d and rate of change of 

j. i dx 

displacement, , 

di f k dx 
/= -*-, 

8 It is, in fact, D'Alembert's principle for the limited case of a single body with a 
single degree of freedom. 



PRINCIPLES OF MECHANICS 13 

where kd is the damping coefficient, usually taken as constant. Such 
a damping force might be exerted by a shock-absorber or dash-pot. 

The negative signs in the above expressions for inertial reaction, 
damping, and resilient forces on a body in a simple system are a con- 
sequence of the physical fact that these forces all oppose its accelera- 
tion, velocity, and displacement, respectively, and that the positive 
direction of force is the same as that of displacement. 

There may be torques on a rigid body in rotation similar to the 
forces on one in translation. The relation between torque and angular 
acceleration may be developed by applying Newton's laws to elemen- 
tary divisions of mass, regarded as particles, and then summing the 
moments of the forces on the particles. Let T denote resultant torque 
applied to the body, / its moment of inertia, 6 its angular displacement, 

d6 . t . , d 2 6 . , f J , 

- its angular velocity, and its angular acceleration, and there is 

dt dt 

obtained 



d 2 x 

which is analogous to / = ma or / = m - . 

dt~ 

T is the resultant torque, in general the sum of torques 7i, 2*2 , . . ., 
or STY In accordance with a generalization of D'Alembert's prin- 
ciple we may write 



d 2 

by including as one torque jT t , the term / , which is the inertial 

dt" 

reaction of rotation. Also, in any system there may be a damping 
device that applies a torque 



and there may be an elastic constraint that applies a torque 

r. = - KO, 

where Kd and K are coefficients taken as constants. Although the 
second relation is quite accurate within limited displacements that 
frequently occur, the former is likely to be only an approximate law, 
of particular value because of its simplicity. 

The above relations may be expressed numerically in any consistent 
system of units. A system of mechanical units commonly employed 



14 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 



by engineers is given in the following table, where, for convenience, are 
also listed the corresponding symbols used in this text. The equations 
that appear later hold in any consistent system of units, and, in par- 
ticular, for the ones of the table. 



QUANTITY 


SYMBOL 


UNIT 


Time 


t 


second 


Distance 


x, y, s 


foot 


Velocity 


V,dx 


foot per second 


Acceleration 


~dl 
a,d*x 


foot per second per second 


Acceleration of gravity 


<5 
g 


32.2 ft. per sec. per sec. 


Force 


f 


pound 


Weight (force of gravity) 


W 


pound 


Mass* 


M 


slue 


Spring constant 


k 


pound per foot 


Damping constant 


kd 


pound per foot per second 


Angular displacement 





radian 


Angular velocity 


o), dO 


radian per second 


Angular acceleration 


di 

a, d*0 


radian per second per second 


Torque 


Jf* 
T 


pound-foot 


Moment of inertia 


I 


slug-foot 2 


Rotational spring constant . . . 


K 


pound-foot per radian 


Rotational damping constant 


K d 


pound-foot per radian per second 









* Mass in slugs = 



Weight in pounds 



Acceleration of gravity, 32.2 ft. per sec. per sec. 



10. Derivation of Differential Equations of Simple Mechanical 
Oscillations. The method described in Chap. I is utilized in deriving 
the differential equations of motion of the following examples, which 
are similar in character to the one of 6 but include more terms. 
These examples will illustrate the application of the principles of the 
previous article. 

(a) Spring, mass, gravity, and damping vane. The configuration 
of Fig. 1, 6, is turned 90 so that motion occurs in the vertical direc- 
tion. That is, a body of mass M is suspended by a spring from a 
fixed support, and is constrained to move only in a vertical line passing 
through the point of support. See Fig. 2. A viscous damping force 
is applied by vanes attached to the body. It is given an initial dis- 
placement below its equilibrium position of rest and is then released. 
Let us determine the differential equation describing the subsequent 



EQUATIONS OF SIMPLE MECHANICAL OSCILLATIONS 



15 



motion when there are no external applied forces except gravity. The 
mass of the spring is assumed to be negligible. 

D'Alembert's principle will be applied to the problem. Accord- 
ingly, the algebraic sum of the forces applied to the body is zero. 
The forces comprise those due to inertial reaction, damping, resilience, 
and gravity. Let x be a coordinate measuring displacement down 




Position with 
spring undistorted 



Equilibrium 

position under 

force of gravity 



FlG. 2. 



A genera! 

position during 

oscillation 



from the position where the spring is undistorted, and let the positive 
direction for force also be down. Then: 

d 2 x 

The inertial reaction is M r^. 

dt 2 

dx 

The damping force is kd (approximately). 

dt 

The resilience force is k x for displacements within the 
elastic limit. 

The gravitational force, equal to the weight, is Mg. 
The equation of motion may then be written, 
d 2 x dx 

dt 2 dt 



(7) 



The equation may be given a simpler form by the choice of a slightly 
different coordinate. Let the displacement be measured down as be- 
fore, but instead of taking the origin as the position where the spring is 
undistorted, let it be the position of equilibrium under the force of 
gravity alone. Let the new coordinate be s, see Fig. 2, and let d 
represent the distance between old and new origins. This distance 
is the displacement required to give sufficient spring force to balance 
the gravitational force. Or 

kd=f a = Mg. (8) 



16 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

The sum of the forces may again be set equal to zero. The forces 
are of the same nature as before and may be written: 

d 2 s 

The inertial reaction = M -77,. 

dt 2 

ds 

The damping force = ka . 

at 

The spring force = k(s + d). 

The gravitational force = Mg. 
Accordingly 

jn j 

-M~-k d ~-ks-kd + Mg = 0. (9) 

dt~ at 

By (8), the last two terms cancel each other, and, after multiplication 
by -1, 

d 2 s ds 



Eq. (10) has one less term than Eq. (7), a simplification which is 
due to the choice of coordinate. It is true in general that if the co- 
ordinate is taken as zero when the body is in equilibrium, instead of 
when the spring is undistorted, the terms representing steady applied 
forces such as gravity disappear from the differential equation. 

Suppose that it is desired to substitute numerical values for the 
constants when the data are given as follows. The weight of the 
body is 20 lb., and under the action of gravity alone, the body has an 
equilibrium displacement of 4 in. from its position corresponding to 
the undistorted spring. The damping force in pounds exerted by the 
vanes is numerically equal to twice the velocity of the body. The 
weight is pulled down 6 in. below its equilibrium position and then 
released. From these data 

*, 20 

^ 



kd = 2 lb. per ft. per sec., 

4 
k. = 20 -T- = 60 lb. per ft. 

X 



Thus the equation may be written 
20 d 2 s ds 



EQUATIONS OF SIMPLE MECHANICAL OSCILLATIONS 17 



The initial displacement which is 0.5 ft. does not enter into the 
differential equation of motion, but it does enter into the solution, as 
explained later. 

" (b) Torques applied to a rotating system. A wheel and shaft of com- 
bined moment of inertia / are mounted in bearings of negligible fric- 
tion with the axis in a horizontal position. A spiral spring mounted 
as shown in Fig. 3 provides an elastic restoring force which opposes 
the angular displacement of the 
wheel. Damping vanes are at- 
tached to it, and also two forces 
/i and /2 are applied tangentially 
to its rim at a distance r from 
the axis, as shown. It is desired 
to determine the differential equa- 
tion of motion of the wheel. 

According to D'Alembert's 
principle, the sum of the torques 
about the axis due to the iner- 
tial reaction, the spring, the 
damping vanes, and the applied 
forces is zero. Let the positive FIG. 3. 

direction for torque and displace- 
ment be counterclockwise, and let 6 be the angular displacement of 
the wheel from the position where the spring is undistorted. Then: 



Damping 
Vanes 



Spiral 
Spring 




The inertial reaction is 



dt 2 ' 

de 



The damping vane torque is Kd -j (approximately). 

at 

The resilience torque is KB, for displacements within 

the elastic limit. 
The torque due to/i is rj\. 
The torque due to/2 is r/2. 

Thus, translating into mathematical form the previous equation in 
words, 



dt 



rf 2 = 0, 



(12) 



which is the desired differential equation of motion. 

(c) Compound system, free vibrations. Forced vibrations of a system 
are those caused by a periodic force applied to it; free vibrations, on 



18 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 



the other hand, are characteristic of the system itself and may occur 
when there is a transient disturbance of its equilibrium. Let us con- 
sider a case of free vibrations of the system shown in Fig. 4, consisting 
of two masses and two resilient members, and constituting a body 
elastically mounted over a single wheel in much the same way, for 
example, that an automobile body is mounted over four wheels. 

Mass Mi is supported by a wheel and elastic tire, and mass Mi 
is supported above M\ by a spring. Constraints not shown permit 
vertical motion only, and the wheel is not allowed to rotate. The 




(a) Spring and 
tire undistorted 



(d) Forced vibrations 
FIG. 4. Coupled Oscillating System. 

masses of all parts except M\ and M% are negligible, as are frictional 
forces. 

Mass M<z is given a downward displacement and then released. 
Until Mz is released Mi is held in the equilibrium position it would have 
if undisturbed. It is desired to determine the differential equation 
describing the subsequent motions of the masses. 

JD'Alembert's principle will be applied to each of the masses. 
Accordingly, the algebraic sum of the forces on each mass is equal to 
zero. The forces on M% include its inertial reaction f Mv the force of 
the spring f sv and the force of gravity f a ^ The forces on Mi include 
its inertial reaction f Mv the force of the spring f Sl , the force of the tire 
f Tv and the force of gravity f Gl . Then 



(13) 



EQUATIONS OF SIMPLE MECHANICAL OSCILLATIONS 19 

Let displacements and forces be taken as positive when directed 
down. In example (a) of this section it was found that simpler equa- 
tions resulted if the origin of displacement was taken at the equi- 
librium position. Therefore take the origins of the displacements Si 
and $2 of masses M\ and Mz at the equilibrium positions. (See Fig. 
4& and c.) The distances of these equilibrium positions from the 
positions of undistorted elastic members will be denoted by d\ and d% 
respectively, as shown in Fig. 4a and b. 

The inertial reactions are 



, 



The spring forces / Sl and f S2 are 



where kz is the elastic coefficient of the spring, and where e is its 
elongation, given by 

e == d\ + s\ d>2 $2- 
Thus 

fsj. = k< 2 (d2 + 52 di Si), 

fs z = ^2(^2 + S2 di si). 

Similarly, if for sufficiently small deformations, the tire conforms to 
the same sort of elastic law as the spring 

/jTi ==- ki(di + si), 

where k\ is the elastic coefficient of the tire. 
The forces of gravity are 



/<* = 

Thus Eq. (13) may be written 

- Mi j + k 2 (d 2 + S2 di - si) - ki(di + s{) + Mig = 0, 

~~ Mz ~T^ 2(^2 + 52 di si) + JkT2|f = 0. 



20 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

These may be simplified by using the relations between the forces of 
gravity on MI and Mi and the equilibrium displacements. The con- 
ditions of equilibrium are that with the bodies at rest, when s\ and $2 
are zero, the sum of the gravitational and spring forces on each body 
is zero. Or 

ki(d% d\) k\d\ + Mig 0, 

(15) 
2(^2 d\) + Mig = 0. 

Eqs. (15) determine d\ and di in terms of the other parameters. Inci- 



dentally, (15) may be written by setting si, $2, -TV ~7T each equal 

at" at* 

to zero in Eq. (14), since these substitutions transform (14), the general 
equations of dynamic equilibrium, to (15), the equations for the par- 
ticular case of static equilibrium. Returning to the simplification of 
(14) we find that certain terms cancel each other as given by (15), 
and leave, 



Mi -- fofe Ji) + kisi = 0, 
at" 

(16) 

d 2 s 2 
M 2 + k 2 (s 2 - Sl ) = 0, 

which are the desired differential equations of motion. 

(d) Compound system, forced vibrations. Suppose now that the 
wheel turns and the whole system has a constant horizontal com- 
ponent of velocity, V. As shown in Fig. 4d the tire runs to the right 
over a road with a series of regular bumps, taken as sinusoidal in 
cross-section. It is desired to determine the differential equations of 
motion on the assumption that the tire remains always in contact 
with the road. 

This case is very similar to the previous one. Again D'Alembert's 
principle is employed, and the same kinds of forces are involved as 
before. 

Since the laws of motion upon which this solution depends hold 
only with respect to a non-accelerated frame of reference, the origins 
of our coordinate system must either be stationary or move with a 
constant velocity. Let us employ the same coordinates s\ and $2 as 
before, permitting the origins to move in the horizontal direction of 
motion with the constant velocity V. 

The expressions for the inertial reactions, the gravitational forces, 
and the force of the spring are exactly the same as before, but that for 
the force f Tl of the tire is different. The compression of the tire is 



EQUATIONS OF SIMPLE MECHANICAL OSCILLATIONS 21 

now di + Si y instead of di + $1, if y is the vertical distance of the 
road surface below its average level. Thus 



f T 



i(di + si - y). 



The quantity y can be expressed as a function of time. Suppose that 
the sinusoidal ridges and troughs have an amplitude y above and 
below the average level, as shown in Fig. 4d, a half wave length L, 
and when time / = 0, y = and the tire is moving downhill. Then 



y = yo sin 



T' 



f Tl = k\ \di 



+ si yo sin 



L r 



Instead of Eq. (16), then, we obtain as differential equations of motion 
for the case at hand 



Mi - - kz(s<2 - $1) + 



dt 2 



i sin 



L ' 



+ k 2 (s 2 - si) = 0. 



(17) 



PROBLEMS 

1. A mass m is free to move along the #-axis. It is acted upon by a force whose 
magnitude is proportional to the distance of the mass from the origin and whose 
direction is away from the origin. Write the differential equation of its motion, 

2. Write the differential equation for the oscillation of a simple pendulum of 
length /, mass m, and angular displacement 6 from 

the vertical. 

3. A uniform circular disc of moment of inertia 
/ is supported by a vertical elastic rod as shown in 
Fig. 5. When the disc is turned about its axis, the 
resilient torque of the rod is k times the angular 
displacement 6 of the disc. The disc is turned 
through an angle and released. Write the 
differential equation of motion of the disc. 

4. A cylindrical buoy floats in fresh water 
with its axis always vertical. The length, radius, 
and weight of the buoy are respectively /, r, and W. 

It is depressed until its upper surface coincides * IG. * 

with the surface of the water and is then released. 

Write the differential equations of motion of the buoy on the assumption that the 

water exerts on it only hydrostatic pressure. 




22 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 



5. A shock-absorber (dash-pot) which acts equally for either direction of motion 

of its piston is placed in parallel with the spring 
of Fig. 4. The force exerted by the shock- 
absorber is always proportional to the difference 
of the velocities of Mi and M 2 . Let M 2 be de- 
pressed while Mi is maintained in its original 
equilibrium position and then both released. 
Write the differential equations of motion. 

6. Write the differential equations of the 
small vibrations of the double pendulum of Fig. 
6. The bobs have masses wi and m*, and the 
strings have lengths a and b. Assume that 
there is no damping. By small vibrations are 
meant oscillations so small that for either 0i or 
2 , sin 6 = 6 and cos = 1. 




FIG. 6.- Double Pendulum. 



11. Solution of Homogeneous Linear 
Differential Equations with Constant 

Coefficients. A homogeneous linear differential equation is one of 
the form 

d n y d n ~ l y dy 

dt n dt n ~ l * * " n ~ l dt 



= 



(18) 



which contains no term independent of y. If p'y, where s 

d*y 
1,2,3, ... w, be written for , Eq. (18) is more simply written 



pny 



a n y = 0. 



(19) 



We shall lead up to the solution of Eq. (19) by first solving some par- 
ticular examples. To solve a differential equation means to obtain 
the general solution. 



EXAMPLE 1. Solve the equation 
P 2 y + aipy + 



= 0. 



(20) 



At this point we encounter a departure from the straightforward 
methods of solution which characterize the techniques of elementary 
mathematics. It becomes necessary to assume a solution and then 
determine its correctness by substitution. Inspection of Eq. (20) 
shows that it might be satisfied by a function y if the derivatives p?y 
and Py were proportional to the function y itself. The derivatives of 
the exponential function y = e mt , where m is a constant, have this 
property, hence we are led to try e m * as a solution, leaving the value 
of m undetermined for the present. Substitution of this value for y 
in Eq. (20) gives 

e mt (m 2 + ci\m + 0,2) = 0. 



HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS 23 

Since e mt cannot be zero for finite values of /, the equation is satisfied 
only if the quantity within parentheses is zero, which means that m 
must have a definite value depending on the constants a\ and 0,2 of 
the differential equation. If it does have this value, e mt is a solution 
of the differential equation (20). The equation 

m 2 + aim + a 2 = (21) 

has two roots, which for some values of a\ and a 2 are real, and for 
others, complex. The assumed solution e mt thus satisfies (20) if m be 
assigned the value of either of the two roots of (21). Moreover, if the 
solution is multiplied by a constant, which may be either real or com- 
plex, the resulting product is still a solution of (20) as may be shown 
by substitution. That is, both y = C\e mit and y = CM* satisfy (20) 
if m\ and W2 are the roots of (21) and C\ and 2 are any real or com- 
plex constants. It may also be shown by substitution that the sum 



y = Cie mit + C*f* (22) 

satisfies the differential equation (20). If C\ and Ci are independent 
arbitrary constants, (22) is the general solution, by the definition of 
8. The two constants in Eq. (22) are independent if they cannot be 
combined into a single equivalent arbitrary constant. Such a com- 
bination can be made only if m\ = mi (See Eq. (30)), and then the 
solution is not the general solution. 

Eq. (21) is called the auxiliary equation or characteristic equation 
of (20). When the roots m\ and m^ of (21) are real, (22) is a convenient 
form for the solution. When the roots are complex, a different form 
is more useful. If the roots are equal, (22) is no longer the general 
solution, and a different form of function must be employed. The 
cases of complex and equal roots of the auxiliary equation will be 
treated in the following examples. 

EXAMPLE 2. Solve the equation 

P 2 y + aipy + a 2 y = 0, (23) 

where the roots of the auxiliary equation m 2 + a\m + a% = are 
complex. If the roots of the auxiliary equation are 4 a bv 1 
= a =t bi, then the general solution of (23), from example 1, is 



y = Cie (a+M)< + C 2 e (a - = e at [CiJ* + C*e- t \. (24) 

Throughout this chapter the symbol i is used to designate V 1. 



24 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

It may be shown as follows that Eq. (24) may be written 

y = e at (A sin bt + B cos bt), (25) 

where only real quantities are present. Since 



Z 3 Z 5 Z 7 

sins = z -- + -_- 

Z 2 Z 4 Z 6 

and 008 * = 1 ~2! + 4!~6l 

for z real or complex, it follows that 



= cos bt + i sin &/. (26) 

Likewise 

c-** = cos&/ - isinbt. (27) 

By the substitution of (26) and (27) in (24), the last becomes 

y = e at [(Ci - C 2 )i sin ^ + (Ci + C 2 ) cos 6/]. (28) 

If y is to be a real quantity, (Ci 2)*' and (Ci + 2) must be real. 

B .4*' 
This implies that Ci and 2 are conjugate. If Ci = - and 

L 

C 2 = - - f w here A and B are real, Eq. (28) reduces to (25), and 

still has two arbitrary independent constants. 

If the roots of the characteristic equation are pure imaginaries 
then a = in Eqs. (25) and (28). Eq. (2) of the introductory prob- 
lem is of this form and can be solved by the method of the present 
section. 

EXAMPLE 3. Solve the equation 

p 2 y - 2 mipy + m^y = 0. (29) 

The auxiliary equation 

m 2 2 m\m + mi 2 = 



HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS 25 

has the double root m = mi. By example 1, the solution is 

y = Cie mit + C 2 e mi< = Ae mit . (30) 

But (30) contains only one arbitrary constant, namely, Ci + 2 = A. 
Thus (30) is not the general solution since by definition the solution 
of a second-order differential equation must contain two independent 
arbitrary constants. However, it can be verified by substitution that 

y = (A + B()e mit 

satisfies the differential equation (29) and, since it contains two arbi- 
trary constants, it is the general solution. 6 

GENERAL EQUATION. We now return to the solution of differential 
equation (19). The principles employed in the last three examples 
are adequate for the solution of this general equation. By substituting 
y = e mt in Eq. (19), the characteristic equation is found to be 

m n + aim"' 1 + . . . + a n = 0. (31) 

If the n roots of Eq. (31) are distinct, then the general solution of 
is 



y = Cie m + C 2 f* + . + CJ*. (32) 

Such complex roots as occur always occur in conjugate pairs. If 
Eq. (31) has only one pair of complex roots (say a hi), then (32) 
may be reduced, by Eqs. (26) and (27), to the form 

y = e at (A sin bt + B cos bt) + C*e m * + . . . + C n f*. (33) 

Similarly, if there are two pairs of complex roots (ai bii) and 
(a2 db b%i), the roots all being distinct, (32) may be reduced to the 
form 

y = e ait (Ai sin bit + BI cos bit) + e a *(A 2 sin b 2 t + B 2 cos b 2 t) 



If r of the n roots of (31) are equal (say Wi = m 2 = . . . = m r ), 
then it is verified by substitution 6 that the general solution is 

y = (Ci + Czt + . . . + C r r V 1 ' +C r +ie m '++ ... + C n e^. (34) 

Finally suppose that one pair of complex roots (a bi) occurs 
r times. Then Eq. (34) becomes 

y = (Ci + C 2 t + . . . + C/- V +M) ' 



"" + . . . + 
* See also 16. See 16. 



26 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

This equation, by means of Eqs. (26) and (27), can be reduced to the 
form 

y = e a \(Ai + Azt + . . . + A r t r ~ l ) sin bt 



+ B 2 t + . . . + B r f- 1 ) cos bt] 

*'^ + .'.. + C n e^ f . (35) 



The equations characterizing free vibrations of a system usually 
are of the form (19), or may be reduced to that form. For example, 
(10), (11), (12), and (16) are such equations. On the other hand, 
forced vibrations, arising in a system to which a periodic force is 
applied, are characterized by non-homogeneous differential equations. 

12. First Method of Solution of Non-homogeneous Linear Differ- 
ential Equations. Reduction by Differentiation to Homogeneous 
Form. A non-homogeneous linear differential equation is one of the 
form 

p n y + aip n ~ l y + . . . + fl-iy + a^y = /(/) (or a constant) (36) 

which thus contains a term independent of y. 

We lead up to the solution of (36) by first solving particular 
examples. 

EXAMPLE 1. Obtain the solution of 

P 2 y + 4py + 3y = sin 2t. (37) 

Write (37) in the form 

P 2 y + 4py + 3y = + sin 2/. 

It is convenient to obtain a solution of (37) consisting of two parts, 

y = yi + , (38) 

where y = y\ is the general solution of 

P 2 y + *py + 3y = 0, (39) 



and y = u satisfies 

P 2 y + tyy + 3;y = sin 2t. (40) 



The functions y\ and u are called respectively the complementary 
function and the particular integral of (37). The complementary 
function contains sufficient arbitrary constants to make the sum 
yi + u the general solution, and the particular integral contains the 
terms on the right side of the equation which represent applied forces. 



FIRST METHOD OF SOLUTION 27 

The auxiliary equation of (40) is 

ro 2 + 4m + 3 = 0, 
which has the roots m = 3, 1. Thus 

yi = Cie-' + C 2 e- 3 '. (41) 

Differentiating (37) twice, there results, 

p*y + p*y + 3p 2 y = - 4 sin 2t. (42) 

If (37) is multiplied by 4 and added to (42), the result is 

p*y + tp*y + 3p 2 y + 4(p*y + 4p y + 3y) = 0. (43) 

This equation is homogeneous. Its auxiliary equation is 

m 2 (m 2 + 4m + 3) + 4(m 2 + 4m + 3) = 0, 
or 

(m 2 + 4m + 3)(m 2 + 4) = 0. (44) 

Attention is called to the fact that if the first factor of (44) is set equal 
to zero we have the auxiliary equation for Eq. (37). The general 
solution of (43) is, from (33), 



y = Cie-* + Cze-*' + A cos 2t + B sin 2t. (45) 

Comparison with (41) shows that (45) contains the complementary 
function. The general solution of (37), since it is of the second order, 
can contain only two arbitrary constants. To determine the value of 
two of the constants in (45), this equation is substituted in (37). 
Since y = Cie~* + C20~ 3 ' satisfies (39), it is time saved to substitute 
only y = u = A cos 2t + B sin 2t in (40). Since 



- 4A cos 2t - 45 sin 2/, 
&4 sin 2t + SB cos 2/, 
and 3y = 3A cos 2t + 3B sin 2t 9 

it follows, on substituting these values in (40), that 

P 2 y + 4y + 3y = - (SA + B) sin 2t + (-4 + SB) cos 2t - sin 2/. (46) 
In order that the equation 

- (A + B) sin 2t + (-A + 8B) cos 2t - sin 2/ = 
shall be true for all values of /, it is necessary that the coefficients of 



28 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

sin It and cos 2/ each equal zero. Accordingly, equating to zero the 
coefficients of sin 2/ and cos 2/ in (46), there results 

- SA - B = 1, 

- A + SB = 0, 
or 



Hence the particular integral of (37) is -^ cos 2t ^ sin 2t, and 
the general solution is 

y = Cie- 1 + C 2 e- 3t - -^ cos 2t - -fa sin 2/. (47) 

EXAMPLE 2. Obtain the general solution of 

P 2 y + 4y + 3y = 4*. (48) 

The second derivative of Eq. (48) is 

ply + p*y + 3p2y = 0. (49) 

The auxiliary equations of (48) and (49) are respectively 

w 2 + 4w + 3 = 
and 

w 2 (w 2 + 4m + 3) = 0. 

The general solution of (49) is 

y = Cie~< + C 2 e-*< + ( 
Substituting y = C 3 + C 4 / in (48), obtain 
4C 4 + 3(C 3 



or 

4C 4 + 3C 3 + (3C 4 - 4)* = 0. (50) 

By equating to zero the coefficients of each power of t 

C 3 = - V , C 4 = 

Hence the particular integral of (48) is / -^, and the general solu- 
tion is 

y = Ci-' + C 2 e- 3 ' + f ^ - y. (51) 

EXAMPLE 3. Solve the equation 

P 2 y + *py + 3y = sin 2t + 4/. (52) 



FIRST METHOD OF SOLUTION 29 

Let the general solution be written 

y = yi + ui + uz, (53) 

where y\ satisfies p 2 y + 4py + 3y = 0, 

u\ satisfies p 2 y + 4/>y + 3y = sin 2/, 
U2 satisfies p 2 y + 4py + 3y 4/. 
From examples 1 and 2 of this article 

yi = Ci-' + C 2 *- 3 ', 

Wi = ^ cos 2/ -g^ sin 2/, 

2=$;-Js* 

Substituting these values in (53) 

y = Cic-' + C 2 *- 3 < - -fa cos 2/ - -fa sin 2/ + f / - V- 

This function y is a solution of (52) because it satisfies the differential 
equation. It is the general solution since it contains precisely two 
arbitrary constants. 

If the right side of (52) had contained n terms, each a different 
elementary function or constant, there would have been n particular 
integrals Wi, W2, . . . u n . 

EXAMPLE 4. (Resonance equation.) Suppose the mass in the intro- 
ductory problem of 6 has a sinusoidal force F cos kit applied to it in 
the line of its motion. If y denotes the same displacement that was 
formerly denoted by x, the differential equation of motion becomes 

k F 

p 2 y + 17 y = 17 cos * i/ - 

M M 
The complementary function is by (25) 

yi=A sin J / + B cos J /, (54) 

^ M ' M 

which describes the free vibrations of the system. The frequency of 

1 [k 

the vibrations is \\ and is called the natural frequency of the 
2ir \ M 

system. The natural frequency in this case depends only upon the 
mass MI and the spring coefficient k. The frequency of the applied 

ki 
force is . When this frequency is equal to the natural frequency 

27T 



30 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 
the system is said to be in resonance with the applied force. 

IT 

Suppose resonance exists, that is, ki = \/T:. The differential equa- 

\M 

tion of motion may then be written 

P 2 y + ki 2 y = ^ cos kit. (55) 

M 

The complementary function is given by (54). The particular integral 
of (55) is found by the method of example 1. Differentiating (55) 
twice 

P*y + ki 2 p 2 y = - ki 2 cos kit. (56) 

M 

Combining (55) and (56) 

P 2 (P 2 y + ki 2 y) = - ki 2 (p 2 y + ki*y), 
or 

(P 2 y + ki 2 y) 2 = 0. (57) 

The general solution of (57), by Eq. (35), is 

y = A i sin kit + Bi cos kit + A*t sin kit + B 2 t cos kit. (58) 
Substituting 



y = Azt sin kit + 2M cos kit 

in (55) and equating to zero the coefficients of like functions of t, we 
have 



If these values of Ai and B% are substituted in (58), the general solu- 
tion of (55) is seen to be 

y - A sin kit + B cos kit + F *** 1 '. (59) 

2kiM 

A MORE GENERAL EQUATION. We now return to the solution of 
the equation 

p n y + aip n ~ l y + . . . + a n -ipy + a n y - /(/), (60) 



where /(/) is a function such that some derivative of /(/) (say the 5th) 
is equal to a constant N times /(/). N may, as in example 2, be zero. 
Differentiate (60) 5 times and obtain 

(61) 



INITIAL CONDITIONS 31 

The elimination of /(/) between (60) and (61) gives 

P + 'y + . + a n p'y - N(py + . . . + ay) = 0. (62) 
The characteristic equation of (62) is 

(m + aim~ l + . . . + On)(W - M) = 0. (63) 



Denote those roots of (63) which are also roots of the characteristic 
equation of (60) by mi, ra2, . . . m. The general solution of (62), all 
roots of (63) being distinct, is 



y = Cie" 11 ' + . . . + CJ + C.+irf*** + . . . + H v? mW . (64) 

The first n terms of the right member of the last equation form the 
complementary function of (60). To determine the particular integral 
u substitute 

y = u = Cn+ie" 1 "* 1 ' + . . . + 



in (60) and equate to zero the coefficient of each exponential term. 
The relations so obtained determine uniquely C n +i, C n +2, . . . d+. 

If some of the roots of the characteristic equation of (62) are 
repeated or are complex then the general solution of (62) is modified 
according to Eqs. (33), (34), or (35). The case of a repeated real 
root is illustrated in example 2, and that of a repeated complex root 
in example 4, of this article. Although /(/) here is somewhat limited 
in nature, it is sufficiently general to include the most frequently 
occurring cases. However, a method of handling Eq. (60) when f(t) 
is a general function of applied mathematics is given in 16. 

13. Initial Conditions. Damping. Harmonic Motion. Let us 
complete the solution of problem (a), (10). Eq. (11) may be written 

33 + 3.22 ^ + 96.6s = 0. (65) 

at* at 

The roots of the characteristic equation of (65) are 

m = - 1.61 9.70, 
and the general solution, by Eq. (33), is 

s = <T L61 ' (A sin 9.70* + B cos 9.70*). 

The last equation may be written, by making use of a trigonometry 
transformation, 

s = Cie~ l 61t sin (9.70* + 0), (66) 



32 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

where C\ and <t> are the new arbitrary constants of the general solution. 
The initial conditions of the problem are 



dt 



when / = 0. 



These conditions give, upon substitution in (66) and the derivative 
of (66), 

% = Ci sin <, 

= Ci(9.70 cos <t> 1.61 sin <), 

9 70 

or = tan- 1 = 80 35', Ci = 0.5068. 

1.61 

Thus the equation of motion of the mass M is 

5 = 0.5068- L61< sin (9.70* + 80 35'). (67) 

In an equation of the form 

5 = Ce~ at sin (ut + 0) (68) 

the angle </> is called the phase angle. (Sometimes epoch angle.) The 
factor e~ M is the damping factor, and a is the damping constant. 
Straight-line motion defined by 

5 = C sin (w/ + <) 

is called simple harmonic motion. The numerical value of C is called 
the amplitude of the motion. The angle <t> is called the phase angle, 

and is the frequency of the harmonic motion. 

As a further example of the evaluation of the arbitrary constants, 
suppose that it is desired to find y in Ex. 1, 12, when the initial condi- 
tions are 

y = 

when / = 0. 



The differential equation is 

P 2 y + py + 3y = sin 2t. 
By (47) its solution is 

y = Cie~* + C 2 e- 3 ' - ^ cos 2t - -fa sin 2*. 



SUMMARY 33 

Substituting the initial conditions into the solution and its derivative, 
we 'obtain 

= Ci + C 2 - eV 

= - Ci - 3C 2 - &. 
From these, 

1 = it 2 = ^-. 
Thus 

' - cos 2 ' ~ -- sin 2/. 



It should be noted that the initial conditions are substituted into 
the general solution and its derivatives, not into only the comple- 
mentary function and its derivatives. 

14. Summary. The steps in solving the homogeneous differential 
equation (19) are : 

(a) Obtain the characteristic equation of Eq. (19) by replacing 
p by m in the differential equation. 

(b) Obtain the roots of the characteristic equation. If this is of 
higher degree than the third, it may be necessary to employ the methods 
of Sec. IV, Chap. II. 

(c) Write the various terms of the general solution recalling that: 
To every non-repeated real root m there corresponds a term of the form 
Ce mt . To every r-fold repeated real root m there corresponds a sum of 
terms 

(Ci + C 2 t + . . . + Crt'-^e"". 

To every non-repeated pair of complex roots a =b bi there corresponds 
the terms 

e at (Asinbt + Bcosbt). 

To every r-fold repeated pair of complex roots a bi there corresponds 
the terms 



t [(Ci + C 2 t + . . . C/" 1 ) sin bt + (/>i + D 2 t + . . . + Z)/" 1 ) cos bf\. 



The quantities A, B, Ci, . . . C r , D\ . . . D r are arbitrary constants. 

The steps in solving the non-homogeneous differential equation 
(36) are: 

(a) Obtain the complementary function by the three steps outlined 
above. 

(b) Find the particular integral by the method of 12 if applica- 
ble, If this method fails, employ 15 or 16, 



34 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 



EXERCISES AND PROBLEMS 

Obtain the solution of the following equations. 

1. P*y - lp*y - py + 7? - 0. 

2. " 1 * 3 * 



3. *? - 2ay 4- *? - 0. 

4. p*y - 2apy + (a* + *)y - 0. 

5. # J y 4y = sin 5J. 

6. p*y - Spy + 2y - e 3 '. 

7. y - (2 + *)# + 2*y - *, 

8. (/> a + W*y - sin R 

9. Obtain the equation of motion in problem 3, 10. The initial conditions are 






when t = 0. 



10. Obtain the equation of motion of problem 4, 10. Give the period and 
amplitude of the harmonic motion. The weight of a cubic foot of water is 62.4 Ib. 

11. In the differential equation obtained as the answer to problem 2, 10, sin 6 
is approximately equal to 6 if is small. Making this approximation obtain the 
equation of motion if the initial conditions are 




dB 



when t = 0. 



15. Second Method of Solution of Non-homogeneous Linear Dif- 
ferential Equations. Operator Method. The method of 12 for 
obtaining the particular integral has the special advantage that it is 
necessary to remember only a process rather than a number of formu- 
las. On the other hand, it has the disadvantage that it is long. 
Furthermore, it is applicable only when pf(f) = constant X/(0 ^ n 
certain cases of very frequent occurrence, the particular integral is 
more quickly obtained by an operator (not operational) method. 

Define the expression 7 



F(p)y m (p + aip*- 1 + + On-ip + a n )y 
by the equation 

(p + aip~ l + . . . + a n )y m py + aip~ l y + . . . + a n y. (69) 
Write Eq. (36) 

TOy-/(0. (70) 

7 Tb symbol is read "is defined tp be," 



SECOND METHOD OF SOLUTION 35 

This equation solved formally for y gives 

y - /0. (7 



The symbol -rrr-r is thus far meaningless. However, y in (70) is a 
F(p) 

function of t such that if operated on by F(p) it gives /(/). In other 

words, rrr-T/(/) is a particular integral of (70). This indicates a 
F(p) 

definition for -rr-r. Accordingly, -zrr is defined to be an operator 
F(p) F(p) 

which is the inverse of F(p), that is 



(72) 



We make use of (72) in obtaining the particular integral in certain 
cases. 

EXAMPLE 1. Find the particular integral of 

F(p) y = e , where F(a) j* 0. (73) 

Now pe at = ae at , 



pn e at = a n e at t 

and 

^ + dip"" 1 + . . . + a)e" = (a* + aia"- 1 + . . . 
or 

F(p)e< = F(a)e ot . (74) 

Applying the operator to Eq. (74) we have, in view of (72), 



~ F(P) ' 
or 

e at e* 



w (7S) 

But from (71), y = - r is the particular integral desired. Thus the 



36 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

gat 

particular integral of (73) is y = u = -ZTT-T. In case F(a) = 0, the 

r(a) 

particular integral is obtained by the method of 12. 
For example, let us obtain the particular integral of 

(p 5 + 6p 4 + 7 3 + 3p 2 



U 



(3) 6 + 6(3) 4 + 7(3) 3 + 3(3) 2 + 11 956* 



This method of obtaining the particular integral of differential 
equations like (73) finds an important application in the determination 
of the steady-state complex number solution of an electric circuit in 
which an alternating voltage is impressed. Thus a circuit differential 
equation might be of the form 

F(P)y = *", 

where y is current or charge. The steady-state solution, which is the 
particular integral, is 

** 



A similar application occurs in the case of a mechanical system to 
which a periodic force is applied. 

EXAMPLE 2. Obtain the particular integral of 

F(p 2 )y = E sin at, F(- a 2 ) ^ 0. 

By a method similar to that employed in example 1 it is easily shown 
that the particular integral is in this case 

E sin at E sin at _ . 

= (76) 



If the function of p on the left-hand side of the equation is not a 
function of p 2 , that is involves p as well, as in the example F(p) = 
p 2 2p + 2, the procedure then is indicated in the solution of the 
following example. Obtain the particular integral of 

(p 2 - 2p + 2)y = E sin 5t. 
E sin 5t E sin 5t 



' p2- 2 p + 2 [p - (1 + *)] [p - (1 - *')]' 

Multiplying the numerator and denominator of this fraction by 



SECOND METHOD OF SOLUTION 37 

[/>.+ (1 + f)] [p + (1 *)]> the denominator becomes a function of 
2. Thus 

_ _ (p 2 + 2p + 2)E sin 5< 

y ~ U ~ [P 2 - (1 + *) 2 ] IP* - (1 - i) 2 ] 
(p 2 + 2p + 2) sin St 



(-25- (1 + ,)'][_ 25 -(1 -*) 



629 - 

And performing the differentiations indicated by p, 

(10 cos S/ - 23 sin 5/)E 



y = u 



629 



In case F( a 2 ) = 0, recourse is had to the method of 12, 
example 4. 

If in (76) sin at is replaced by cos at, a solution may be obtained 
by a similar method. See 17. 

EXAMPLE 3. Obtain the particular integral of 
(P 2 + 2p + 2)y = E sin 5/ + Fe*<. 
E sin St Fe 3t 



p* + 2p + 2 p* + 2p + 2 

_ (10 cos 5* - 23sin5Q F& 
629 + 17 

(by examples 1 and 2). 

EXAMPLE 4. Methods of proving the following formulas are given 
in 123. 

(77) 



where F\(f) is any function of / and F'(p) 



' <78) 

dF(p) 




. 
dp 

These relations are used as follows. In formula (77) let 
a* = - 1, Fi(t) = E sin St, F(p) = p 2 + 1. 



38 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 
Then 



r- sin5 * 
U> 2 -2*> 

and by example 2, 



e~' [10 cos 51 - 23 sin 5<] 



629 
In formula (78) let 



2 + 2 + 2, 
Then 



, e = I", _ (^ + 2) 1 1 ,. 

L 2 + 2 + 2j 2 + 2 + 2 



5 5(/> 2 + 2/> + 2) 

te> 2(p + l)e'1 
T -- 2l~J 



16. General Method of Solution of Non-homogeneous Linear Dif- 
ferential Equations. A general method is sometimes necessary which 

is applicable to every /(/) for which I ... I e at f(t)dt n exists. In 

*/o /o 
developing a general method, we first establish the relation 

+ ...+ a n )y = (p - mi)(p - m 2 ) . . . (p - m n )y f (79) 



where the left side is defined by (69) and where wi, W2, . . ., m n are 
the roots (real or complex) of 

m n + aim n ~ l + . . . + a n _iw + a n = 0. (80) 



If p were merely an algebraic symbol instead of a differential operator, 
Eq. (79) would follow immediately from the factor theorem of algebra. 

However, it is not difficult to establish Eq. (79) for p = -7. AH 

at 



GENERAL METHOD OF SOLUTION 39 

essentials of the proof for the general case are included in the proof of 
the special case for n = 2, that is 

(P 2 + aip + a 2 )y = (p - mi)(p - m 2 )y. (81) 



That (79) is true for w = 2 is easily established as follows. If n = 1 
in Eq. (69), there results 

(P - mi)y = py - m\y. 
If (p mi)y be denoted by v t then by (69) 

(p m 2 )v = pv m 2 v, 
or 

(p - w 2 ) [(p - mi)y] = p(p - mi) - w 2 ( - mi) 



= p 2 (mi + m 2 )p + wiw 2 . 

But the relations between the roots mi and m 2 and coefficients a\ and 
a 2 of the quadratic equation p 2 + ai/> + a 2 == are 

~ (mi + w 2 ) = ai, miw 2 = a 2 . 
Thus 

(p - 



By the same argument, it is evident that 

(p - m 2 )(p - mi)y = (p 2 + aip + a 2 )y, 



where the order of the factors in the left member is changed. 

By obvious extensions of the proof for n = 2, the general equation 
(79) can be proved and by such proof it can be shown that the order 
of the factors in the right-hand side of (70) is immaterial. 

The method of obtaining the general solution of 

(p - mi)(p - 1112) ...(/>- m n )y = /(/) (82) 

is clearly explained by consideration of the special case 

(p - mi)(p - m 2 )y = /(/). (83) 

Let 

(p - m 2 )y = v. (84) 

Then (83) is 

(p-mjv =/(/). (85) 

If Eq. (85) is multiplied by e~ mit and both sides of the equation inte- 
grated with respect to /, there is obtained 



40 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

or 

v = e mit I e' mit f(t)dt + Cie mit . (86) 



Substituting this value of v in (84), we have the non-homogeneous 
equation 

(p - m 2 )y = e mit J e~ m f(t)dt + Cie mit . (87) 

Multiplying (87) by e"" 2 ' and integrating both sides with respect to t 
we obtain 



- Ci \dt + C 2 , 
J LJ J 

or 

r r r i 

JMlt i -(Wi Wfl)/| Wl\lr/M\ji i /" 1,7^ I /" JMzt /OO\ 

y e I e \ I e J w* 4~ <^i <** ~r ^2^ w) 

^ IJ J 

Eq. (88) is the general solution of (83). However, it is easier to 
obtain the complementary function by the method of 14 and to 
employ (88) to find only the particular integral. Hence, to obtain 
the particular integral the arbitrary constants C\ and Cz, in the solu- 
tion just obtained, must be set equal to zero. 

If Eq. (81) had contained n factors instead of two, a continuation 
of the process outlined in Eqs. (84-88) would yield either the par- 
ticular integral or the general solution according as Ci, . . . . C n were 
pr were not zero. 

This general method holds whether the roots mi, W2, . . . m n are 
real or complex. 

EXAMPLE. Obtain the general solution of (p 2 + l);y = / by the 
method of 16. 

In (p i)(p + i)y = t, denote (p + i)y by v. We desire first the 
solution of 

(p i)v = t. 
By Eq. (86) 

v = it+ 1 + Ae { \ 

where A is an arbitrary constant. Substituting the value of v just 
obtained in (p + i)y = v, the non-homogeneous equation 

(P + i)y = */+!+ Ae 

is obtained. Multiplying this equation by e {i and integrating both 
sides with respect to /, we have 



SUMMARY 41 



or 

A 

y = - e" + Cze-" + 



= Cie il + Coe-" + t = C 3 sin * + C 4 cos * + t. 

This is the general solution sought. 

In general, the evaluation of the integrals in Eq. (88) will not yield 
a finite sum of elementary functions. On the other hand, if the inte- 
grands are expanded in series and the integrations performed on the 
series the functional properties of the solution are obscured. Con- 
sequently, it is frequently preferable, in an engineering problem, to 
expand the /(/) of Eq. (82) in a Fourier series (see Sec. Ill) and 

obtain the particular integral by the application of - to the first 

F(p) 

few terms of the series as explained in the previous article. 

17. Summary. The general method of 16 is theoretically appli- 
cable to every Eq. (60), where f(t) is any applied force or voltage which 
is a function of the time. However, if f(t] is not one of the special 
forms treated in 12 and 15 it is preferable, from an engineering 
standpoint, to solve the equation by the method described in the last 
paragraph of 16. 

If fit) is one of, or a linear combination of, the forms e at , sin at, 
cos at, t Fi(t), e at Fi(t), then the particular integral may be written 
down at once by means of the following formulas: 



(89) 



sin a* , . 

(90) 



1 cos at 

cos <" = 



(92) 

(93) 



42 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 



where <|>(^) is a function such that F(p) $(p) * f(p 2 ) is a function of p 2 . 

(94) 



The complementary function in every case is obtained by the 
method of 14. 

EXERCISES AND PROBLEMS 

1. By the method of 16, obtain the general solution of (p a)*y = 0. 

2. Solve (p* + l)y = te*. 

3. Solve (/>* + a*)x - 2D cos w/ + 0.5D cos 2w* + E. 

Hint: Let # = y + k and the differential equation becomes (/> 2 + fl 1 )? = 2D cos 



E 

o>/ + 0.5D cos 2w/ -h E a 2 . If jfe = the differential equation in y contains no 

a* 

constant term. 

4. Solve (p* -f 4)y sin 3t -f- cos 3/. 

5. Solve I p* + 7 # -f 77 jy - <*i sin / -f a sin 2/ + a s sin 3t + bi cos / 

4- 62 cos 2* + 6, cos 3*, 
where the only variables are y and i. 

The next three exercises are concerned with the solution of three frequently 
occurring types of differential equations whose coefficients are not necessarily 
constants. 

6. Find the general solution of the first order linear differential equation, 

dy 

+ Py = Q t where P and Q are functions of /. 



dt 



/Pdt 
we have 



Integrating with respect to t 



(It is easily verified by differentiation that e* y is the integral of the left side of the 
differential equation.) Finally 



This is the general solution since it satisfies the differential equation and contains 
one arbitrary constant. 



SUMMARY 43 

7. Obtain the general solution of 

x(l + y*)dx + y(l + x*)dy - 0. 
This type is known as variables separable. Separating the variables, the equation is 

* dx , y*y ^o 

1 + x t T i 4. y j 
Integrating 

i log (1 + *) + i log (1 + y 1 ) - Ci - log C, 



<J J y dy 
8. Obtain the solution of J* -~j < ~ + 4y /*. 

An equation of this form is called a homogeneous linear differential equation. By 
means of the substitution / c* it can be reduced to a linear differential equation with 
constant coefficients. Let / e x . Then 



dx' 
and 



dt \ dx 



Substituting these values of the derivatives in the original equation, it reduces to 

S-2+*-" 

9. Solve the equations: 

(a) 3c* tan ydx + (1 - /) sec 1 ydy - 0, 



(c) sin x cos ydx cos x sin ydy 0. 
10. Solve the equations: 



(c) 



44 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

11. Solve the equations: 

(a) t*p*y + 3tpy + y = 0, 

(b) Pp*y + 2tpy - 6y = 3/ 2 + 4, 

-f 3//>? + y = sin /. 



18. Simultaneous Linear Differential Equations. Methods of 
solving simultaneous linear differential equations with constant coeffi- 
cients are illustrated by the solution of some typical examples. 

EXAMPLE 1. Solve the system of differential equations 



-* + 4, =0, 

or 

(p + 2)* - 3y = 0, (96) 

- x + (p + 4)? = 0. (97) 

If (97) is multiplied by (p + 2) and added to (96), the equation 
(p 2 + 6p + 5)y = is obtained. The general solution of this equa- 
tion is 

y = Cie- 1 + Cze- 5t . (98) 

Substituting this value of y in (97), the value of x is 

x = 3Ci*-< - C 2 e- 5t . (99) 



The system of Eqs. (98-99) is the general solution of the system of 
differential equations (96-97). 

The order of a system of linear differential equations is in general 
the sum of the orders of the highest derivatives appearing in each of 
the differential equations. The order of the above system is 2. Con- 
sequently, the number of arbitrary constants in (98) and (99) is 2. 

If the value of y in Eq. (98) is substituted in (96), the solution of 
the resulting equation for x gives 

x = C 3 e- 2 ' + 3Cie-< - C 2 e- 5t . 

This value for x and the value of y in (98) will satisfy (97) only in 
case Cs = 0. Thus, it is immaterial in which of the two differential 
equations the value of y is substituted; the solution is the same. 



SIMULTANEOUS LINEAR DIFFERENTIAL EQUATIONS 45 

EXAMPLE 2. Solve the system of differential equations 

(p + an)* + 012? = e 3 ' (100) 

021* + (p + a 22 )y = 0. (101) 

Multiplying (100) by 021 and (101) by (p + an) and adding the 
two equations, we have 

[(p + aii)(P + #22) 012021] y = 02i 3 '. (102) 

Multiplying (100) by (p + 022) and (101) by 012 and adding, we have 

[(p + a 22 )(p + a n ) - 012021] * = (3 + a 22 )* 3 '. (103) 

Let the distinct roots of (m + 022) (m + an) ai202i = be mi and 
m 2 . The general solutions of (102) and (103), respectively, are 

y = Cif* + C 2 e^ - ^ e*<, (104) 



x = CV* + CV* + (3 + ! 22)e3 ', (105) 

1 

where 

Z = (3 + 022) (3 + 0n) - 012021- 

Since the general solution of the system (100-101) will contain only 
two arbitrary constants, relations exist between C'\ and C\ and between 
C' 2 and C 2 . If (104) and (105) are substituted in either (100) or 
(101) and the coefficients of e mit and e m<it each set equal to zero, the 
relations between the constants are found to be 



C\(mi + an) = - 012(^1, C' 2 (m 2 + 011) = 0126*2 

or their equivalents. If C\ and C'z are eliminated from (105) by 
means of the last equations, then (104-105) are the general solution 
of (100-101), provided (p + an)(P + 022) 012021 is not equal to 
zero for every value of p. If (p + a\\)(p + 022) 012021 is identically 
zero, then (100-101) have no solution. Whether a system of simul- 
taneous linear differential equations with constant coefficients has one, 
none, or an infinitude of solutions depends upon the coefficients of the 
dependent variables and the non-homogeneous terms or the right- 
hand side of the differential equations. The reason for this will be 
made clear in Sec. II. 

EXAMPLE 3. Solve Eqs. (17) of 10, that is 

(Mip 2 + ki + k 2 )si - k 2 s 2 = kiyo sin ~ , (106) 

jLtf 

= 0. (107) 



46 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 
Eliminate 52 between (106) and (107) and obtain 

(Mik 2 + M 2 ki + M 2 k 2 )p 2 + kik 2 ] si 

= kiyo(k 2 - M 2 a 2 ) sin at, (108) 



where a = . The roots of the characteristic equation 
L 



(Mik 2 + M 2 ki + M 2 k 2 )m 2 + kik 2 (108a) 
are =t wit, =b a>2t, where 



-(Mik 2 +M 2 ki+M 2 k 2 )+V(Mik 2 +M 2 ki+M 2 k 2 ) 2 - 

2MiM 2 



2MiM 2 
If wi j W2, then the general solution of (108) is 

$1 = Ci sin wi/ + 2 cos w i^ + 3 s ^ n W2 ^ + *4 cos W 2/ 

- M 2 a 2 ) . 
- - sin at, (109) 



where Z = JkfiM 2 a 4 - (Mi* 2 + M 2 ki + M 2 k 2 )a 2 + kik 2 . 

Substituting (109) in (107) and solving the resulting differential equa- 
tion for s 2t we have 



S 2 = 5 sin at + CQ cos at 



- , , , , . , , 

sin wi/ + ~ cos wi/ H -- sin W2/ H 

)3 7 7 



C \ 

COS W2/ I 

7 / 

, J\)r*l.\r*'& " Jyi 2 a ) , t**r\\ 

+ ^ sm a/, (110) 



u * 

where a = 



ff == k 2 

7 = k 2 
- k 2 - 



ELECTRIC CIRCUIT PRINCIPLES 47 

If (110) is substituted in (106) and the coefficients of sin at and cos at 
are set equal to zero, Cs = Ce = 0. Finally, the general solution of 
(106-107) is 



s\ = Ci sin o>i/ + 2 cos o>i/ + 3 sin o>2/ + C* cos 2* + Fsin at (111) 

2 = C'i sin wi/ + '2 cos o>i/ 

JF 

+ C' 3 sin o>2/ + C"4 cos w 2 / + sin at, (112) 

Zi 

where <7, - *', CT, - ^, * = *' y ( * 2 " 



4 = 



Let the spring and tire system represented by Eqs. (106-107) be 
the wheel and tire represented in Fig. 4c of an automobile such that 
the constants of the equations are 

Mi = 3.105 slugs, M 2 = 31.05 slugs, Jfe 2 = 3000 Ib. per ft. 
ki = 13,200 Ib. per ft., and L = 1 ft. 

Let us find the two speeds of the automobile which will cause resonance 
in the spring and tire system. 

If the numerical values for k\ 9 2, Mi, and MT, are substituted in 
the formulas for o>i and o>2, the values for coi and 2, respectively, are 

* irV wV 

4.40 and 72.77. Setting = o>i and -r- = o>2, the two values for 

Lt Li 

V are, respectively, 1.40 and 23.16 ft. per sec. (0.95 and 15.79 miles 
per hour). 

Methods of obtaining the particular integral of the differential 
equations of mechanical systems having many degrees of freedom or 
of circuits with many meshes are given in Chap. II, Sec. II. Oper- 
ational methods of solving such systems of differential equations are 
given in Chap. IV. Analogies between mechanical and electrical 
systems are discussed in the paper of Ref. 10 at the end of the text. 

19. Electric Circuit Principles. The differential equations for 
electric circuits with lumped parameters are of exactly the same form 
as the equations for mechanical systems that were derived and solved 
in the preceding work. Kirchhoff s electromotive force law plays the 
same r61e in setting up the former equations as D'Alembert's principle 
does in setting up the latter. Kirchhoff 's laws may be stated : 



48 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

1. The algebraic sum of the electromotive forces around a 
closed circuit is zero. 

2. The algebraic sum of all the currents into the junction point 
of a network is zero. 

The first law can be shown to be an application of the law of con- 
servation of energy to a circuit; the second, a statement of the 
conservation of electricity. The algebraic sign of an electromotive 
force or of a current indicates its direction. It is implied therefore 
that a positive direction with respect to the current must be specified 
arbitrarily in order that the symbols representing electromotive force 
and current may have physical significance. 

In addition to electromotive forces applied externally, for example, 
by batteries and generators, there are electromotive forces due to the 
current in the circuit elements. If the positive direction for electro- 
motive force is chosen the same as that for current, these are: 

di 

Electromotive force of self-inductance = L , 

at 

Electromotive force of resistance = Ri t 

Electromotive force of capacitance = , 

O 

where the coefficient of self-inductance L is in henrys, the resistance 
R in ohms, the capacitance C in farads, the current i in amperes, the 
charge q in coulombs, the electromotive force e in volts, and time t in 
seconds. Current and charge are related by the equation 

. dq 



A comparison with the mechanical principles shows that electric 
charge is analogous to mechanical displacement, current to velocity, 
electromotive force to force, inductance to mass, resistance to viscous 
damping, and reciprocal of capacitance (1/C) to the spring coefficient. 

If two circuits are coupled magnetically the electromotive force of 
mutual inductance in the first due to a change of current in the 
second is , . 



and similarly, the electromotive force in the second due to a change 

of current in the first is , . 

r di\ 

62 = L,2\ , 
at 



DIFFERENTIAL EQUATIONS OF SIMPLE LINEAR CIRCUITS 49 

where L\i and 21 are positive constants provided that currents flowing 
in the positive directions in both circuits produce magnetic fluxes that 
link either circuit in the same direction. If the fluxes due to positive 
currents link either circuit in the opposite directions the algebraic 
signs of the electromotive forces of mutual induction are changed. 
It can be shown that the mutual inductances Li2 and L^\ are equal. 
20. Derivation of Differential Equations of Simple Linear Circuits. 
The following differential equations are derived by means of the prin- 
ciples of 19. The symbols 



r OW VW\r 



L r*~ R 



have the same significance as 
given there. 

(a) Simple series circuit. It 
is desired to determine the dif- 
ferential equation of the simple 
series circuit shown in Fig. 7, 
consisting of an inductance, FIG. 7. Simple Series Circuit. 

resistance, and capacitance in 
series with a battery of constant electromotive force E. 

According to Kirchhoff's first law the algebraic sum of the electro- 
motive forces around the circuit is zero. Let the positive direction 
for current and voltage be chosen as clockwise. The electromotive 

di 
forces in that direction are then: L due to the inductance, Ri 

at 

due to the resistance, due to the condenser, and E due to the 

o 

battery. Thus the equation may be written: 

-Lj-Ri-^ + E = 0. (113) 

at L 

Or, by using the relation between current and charge on the con- 
denser, (113) may be rewritten, after changing signs throughout, 



(b) Circuits with conductive coupling. Let us determine the dif- 
ferential equations for the circuit shown in Fig. 8, which contains an 
alternating voltage E sin w/. 

According to the first of Kirchhoff's laws the sums of the electro- 
motive forces around the closed circuits / a b e and b c d e must each 
be zero. 



50 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

Designate the currents in the three branches by ii, 2, and iz as 
shown in the figure, and let the associated arrows indicate the positive 
direction for both current and electromotive forces in the respective 
branches. Summing the electromotive forces around the first circuit 
we obtain 

- Li -p - Riii - TT - 12*3 + E sin w/ = 0. 
at Ci 

Around the second circuit 

(7o dlf2 

- ~ - R 2 i 2 - L 2 + Rvtia = 0. 
C2 at 



UUV VVVV II 
L, g, f 

Eslnwt 1 < 


1| VVVV VVV 
Cg 

> 
> 

> 



f e d 

FIG. 8. 

By the second of KirchhofFs laws, 

ii i2 is = or is = ii {% 

Using the last relation to eliminate is from the first two equations, 
and rewriting, we obtain 

1 -T? + (Ri +"12)11 + TT - 12*2 == sin wf, 
a/ Ci 

L 2 ^7 + (2 + 12)12 + ^ - IZurfi = 0. 



Expressing the currents in terms of charges, 



, ^ 

+ - 12 - - 0. 



(114) 



DIFFERENTIAL EQUATIONS OF SIMPLE LINEAR CIRCUITS 51 



(c) Circuits with condensive coupling. Let us obtain the differential 
equations for the currents in the condensively coupled circuits of 



c, 



-^TTBTT^ 



FIG. 9. 

Fig. 9. By the same procedure as used in the previous example the 
following equations are found: 



dt 2 dt 



(115) 




FIG. 10. 

(d) Circuits with transformer coupling. Let us obtain the differ- 
ential equations for the currents in the inductively coupled circuits of 
Fig. 10. 

By the first of KirchhofFs laws the sum of the electromotive forces 
around each circuit is zero. Let the positive direction for current 
and voltage be the same in each circuit, and let the positive directions 
for the two circuits be so related that positive currents cause fluxes in 
the same direction in the magnetic circuit. In the first circuit the 

electromotive forces include: E sin w/ due to the generator, ~- due 
to the condenser, R\ii due to the resistance, 1 -37 due to self- 



52 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

induction, and L\2 due to mutual inductance. Thus the equa- 
dt 

tion of the first circuit becomes 



dt Ci 

Similarly, that of the second is 



dt ' "*" ' C 2 ' ~" dt ' 
Using the relation between currents and charges, we obtain 



at 
dq2 



n 
= 0. 



(116) 



For the set-up of differential equations of motion of complicated 
electrical, mechanical, or electro-mechanical systems see Ref. 7 at end 
of text. 

PROBLEMS 

1. The differential equations obtained in problem 5, 10, are 



dp \dt dt) 



- Si) = 0, 



Obtain the general solution of these simultaneous equations. 
2. The differential equations obtained in problem 6, 10, are 



4- 6m 2 + (mi 



Bi = 0, 



FIG. 11, 



Obtain the general solution of these simul- 
taneous equations. 

3. Find the charge on the condenser in 
terms of time / in the network represented 
in Fig. 11 if the current through inductance 
and all charges are zero at time / =0. 

4. Find the currents in each branch and 
the charge on the condensers in terms of 



DIFFERENTIAL EQUATIONS OF SIMPLE LINEAR CIRCUITS S3 



time in the network represented in Fig. 12 if all currents and charges are zero at time 
t = 0. 



Ri 

-AA/Y 



Rt 

AA/V 



FIG. 12. 
5. Write the differential equations for the circuit of Fig. 13. 




6. Obtain an equivalent electrical circuit for the differential equations of prob- 
lem 1. 

7. A uniform beam of weight W and length / is hinged at B and supported in a 
horizontal position at A by a spring as shown in Fig. 14. The spring constant is k. 



'////////////////////////////////////////////////s///////////// 
FIG. 14. 

The left end of the beam is depressed slightly and suddenly released. Assuming 
that the beam is rigid, find the differential equation of motion and the period of 
oscillation. 

8. A uniform beam of mass M and length 21 is supported on two springs $1 and $2, 
as shown in Fig. 15, and such that the beam has but two degrees of freedom; one an 
oscillation of the center of gravity in a vertical line, and the other a rotation about 
a line through the center of gravity and perpendicular to the plane of the figure. 



54 DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 

4 

Find the equations of motion and the periods of oscillation for free vibrations of the 
system. 

9. An inextensible string is coiled around a rough circular homogeneous cylinder 
of mass M and radius r. One end of the string is attached to a stationary point of a 
horizontal plane such that when the cylinder is rolled up the string and touching the 





M 










1 


o ; 





1 





o 


( 





o 


1 
( 


s* 


s* 


1 





o 


' 








I 


( 










FIG. 15. 



FIG. 16. 



plane the string is vertical at its first point of tangency to the cylinder as shown in 
Fig. 16. The cylinder is dropped. Write the differential equations of motion 
assuming that the axis of the cylinder is constrained so that it remains horizontal. 

10. A body of mass M starts from rest on the rim of a hemispherical bowl of 
radius r and slides down the inside of the bowl under the influence of gravity. The 
friction force acts tangentially to the surface and is proportional to the normal force 
between the weight and the surface. Write the differential equation of the motion 
of the body, assuming that its dimensions are small compared to the radius r. 

11. A pulley has a radius of 1 ft., radius of gyration 6 in., and weight 200 Ib. A 
rope passing over the pulley is attached to a weight of 90 Ib. on one side and to a 
spring on the other. The constant of the spring is 10 Ib. per in. of deflection. The 
system is initially at rest with a 2-in. deflection in the spring, and is then allowed to 
move under gravity. Obtain the equation of motion and the period of oscillation. 

12. Some passenger elevators have been equipped with an air-cushion safety 
device intended to bring the elevator to a safe stop in case of a free fall. The car is 
made to fit the shaft closely, thus acting as a piston in a cylinder. This close fit 
exists only near the bottom of the shaft. 

An elevator 5 ft. square weighing 3 tons was traveling upward at the rate of 800 ft. 
per min. When it reached a height of 700 ft. above the ground floor, the cables 
broke. The car came to a stop and then fell freely to a point 100 ft. above the 
ground floor, where the air-cushion safety device began. Assuming adiabatic 8 com- 
pression of the air, find the position of the car at any time J. 



8 When air is adiabatically compressed the pressure and volume are related by 
the equation PV 1 '* constant. 



INTRODUCTORY PROBLEM 55 



DETERMINANTS 

The following are a few of the numerous engineering applications 
of determinants. Determinants are advantageously employed in solv- 
ing linear homogeneous and non-homogeneous algebraic equations, 
and in giving a criterion for independence of linear algebraic equations. 
Linear homogeneous and non-homogeneous algebraic equations may 
arise, for example, in the solution of simultaneous differential equations 
with constant coefficients. The proof of Bromwich's fundamental 
theorem of the operational calculus makes extensive use of determi- 
nants. The criterion for stability of electrical and mechanical sys- 
tems, whose differential equations are linear with constant coefficients, 
is most conveniently expressed in determinant form. The applications 
of dyadics in synchronous-machine theory and in the theory of elas- 
ticity are frequently made in determinant form. The study of equiv- 
alent circuits is facilitated by use of determinants. This section is 
concerned with a brief introduction to some of the important proper- 
ties and theorems of determinants and with some of the above appli- 
cations. Most of the applications, however, occur later in the text. 

21. Introductory Problem. Before considering the properties of 
determinants, let us see in a simple example how they may be used in 
the solution of algebraic equations. Suppose that it is desired to solve 
the two linear equations 

^ + ^ = *" 1 (117) 



If we multiply the first equation of (117) by 62 and the second by b\ 
and add, we have 



The binomial expression (aife #2*1) may be represented by the 
symbol 

01 1 < 119 > 



Later the binomial will be seen, by definition, to be a determinant of 
second order. (Sometimes the symbol itself is called a determinant.) 



56 



DETERMINANTS 



Using a symbol similar to (119) to designate (fei&2 &2&i), we may 
write as the solution of Eq. (118), 

h 



x = 



01 hi 

02 l>2 



(120) 



Similarly, multiplying Eqs. (117) by #2 and 0i and adding, the 
solution for y becomes 



y = 



(12 



(121) 



Eqs. (120) and (121) express in symbols the solution of Eqs. (117) 
as the quotients of two binomials or determinants. The solution of a 
system of n non-homogeneous equations in n unknowns may be 
expressed just as simply in the symbolism. If n is greater than 2, the 
method ordinarily employed in solving the equations becomes labori- 
ous, and the use of determinants may effect a saving of time and labor. 
The procedure is essentially to express the solution as the quotients of 
determinants, and then to evaluate them by algebraic operations. 
This leads us (from the engineering viewpoint) to the study of the 
relations between the determinant, which is a polynomial, and its 
symbol, which is a square array of letters or numbers called elements 
of the determinant. We desire to know how to expand the symbol 
into the polynomial it represents, how to simplify it to make the 
expansion easier, and the effect upon the value of the determinant of 
certain operations on its symbol. Lack of space prevents the inclusion 
of the proofs of most of the theorems given. 

22. Definitions. Determinants of the fourth order and of the nth 
order are denoted respectively by the symbols: 



0n ai2 013 014 

021 022 023 024 

031 032 033 034 

041 042 043 044 



and 



011 012 .... 01n 
021 022 .... 02n 



0nl- 



0nn 



Abbreviations for these two symbols are respectively | #44 [ and | nn |. 



DEFINITIONS 



57 



An nth order determinant is a certain homogeneous polynomial 9 of 
the nth degree in the n 2 elements #/, where i and j represent integers 
from 1 to n inclusive. The explicit form of this polynomial is given 
in the next paragraph. The elements may be constants or variables, 
and in the general case when they are not given numerically, each 
element is designated by a double subscript which indicates the row 
and column in which the element may be found. For example, 042 
is the element from the fourth row and the second column. The 
symbol of the nth order determinant is composed of n horizontal rows 
and n vertical columns of the elements. A determinant of order 
n 1 may be formed from a determinant of order n by striking out 
or erasing any row, say the ith, and any column, say the jth, inter- 
secting at the element a,-/. Such a determinant of order n 1 is 
called the minor (strictly speaking, the first minor) of a.,-. This minor 
is denoted by Af,-/ f where the subscripts are the same as those of the 
element common to the struck-out column and row. For example, in 
the fourth-order determinant given above, the minor of 023 is 



0n 

031 
041 



012 
032 
042 



014 
034 
044 



The explicit form of the polynomials referred to in the preceding 
paragraph is as follows: 



022 



011 012 
021 022 



= 011022 01202L 



Similarly, 



033 ss 



011 012 013 
021 022 023 
031 032 033 



011022033 + 012023031 

:+ 013021032 011023032 

012021033 ~ 013022031. 



That is, the polynomial is the sum of all the different products that 
can be formed from the symbol by taking one element from each row 
and one element from each column, the sign of the product depending 
upon the way elements are chosen. For a fourth-order determinant 

I 044 I = S rt 01g02r0304, 

where g, r, s, t is any one of the 24 permutations of 1, 2, 3, 4. The sign 
of each term is + or according as an even or odd number of inter- 

9 A homogeneous polynomial is one all of whose terms are of the same order in the 
variables; e.g., x* + I3x*y* 4xy 3 -f yx* -f y* is homogeneous of the fourth order in 
x and y. 



58 



DETERMINANTS 



changes is necessary to derive the arrangement g, r, $, / from 1, 2, 3, 4. 
To illustrate 

an 012 



021 022 



#11022 ~ 



Here 011022 has the plus sign since the order of the second subscripts is 
1, 2. The second term has the minus sign since one interchange is 
necessary to bring 2, 1 into the order 1, 2. 

Or, in the above example of the third-order determinant, we have 
for each term of the right member the order of the second subscripts 
given in the following table. The number of interchanges necessary 
to put it in the order 123 is given; if this number is even, the sign of 
the term is positive, if odd it is negative. 



TERM 


ORDER 


No. OF INTERCHANGES 
TO PUT IN ORDER 123 


SIGN 


First term 


123 





-f 


Second term 


231 


2 


T- 


Third term 


312 


2 


-f- 


Fourth term 


132 


1 




Fifth term 


213 


1 




Sixth term 


321 


3 













Finally, the polynomial form of the nth order determinant is 



nn = - 



0nt n , 



where ii, 1*2, t'a, ... i is an arrangement of 1, 2, ... derived from 
1, 2, ... n by i interchanges. 

The definition of a determinant has no direct application in engi- 
neering, but theorems for the evaluation and manipulation of deter- 
minants are proved directly from it. 

23. Laplace's Expansion. Laplace's expansion is a convenient 
method of finding the polynomial corresponding to a given symbol. 
The rule for Laplace's expansion of a determinant of the nth order is: 

(0) Form the n products a^M /, where either i or j is fixed while 
the other takes the values 1 to n. This corresponds to finding all the 
products, along any one column or row, of each element by its minor. 

As an example let us expand the third-order determinant 

011 012 013 
021 022 023 
031 032 033 



LAPLACE'S EXPANSION 



59 



along the second row. Then i = 2, j = 1, 2, 3. The three products 
are: 

021-^21, 022-&f22 ctnd 023-M23. 

(6) Attach to each product its sign as determined from the checker- 
board array where the sign attached to a\iM\\ is always plus. This 
array of signs corresponds to the array of elements in the symbol 
(third- and fourth-order determinants) : 



(or give 0,-/Af/ the sign ( I)*"*"', which is the same thing). 

The signs of the above products thus are 02iAf2i, + 022^22, 

023-^23* 

(c) Form the algebraic sum of the n products. We have now 
expressed the nth order determinant as the algebraic sum of n deter- 
minants of order n 1. 

The algebraic sum of the products in the example is 

A = 021-M21 + 022-M22 023-M23. 

(d) Continue to apply steps (a), (6), and (c) to the second, third, 
etc., minors until MH is of order one. 

Expanding M^i by its first row 



Similarly 



032 033 

011 013 

031 033 

011 012 

031 032 



= 012033 013032. 



= 011033 013031f 



011032 ~ 012031. 



Thus 



A = 



021012033 + 021013032 + 022011033 

022013031 023011032 + 023012031i 



which is equal to the previous definition for the third-order determi- 
nant although the arrangement of terms and factors is not the same. 

The proof of Laplace's expansion for a determinant of the third 
order may be completed by making the other five possible expansions 



60 



DETERMINANTS 



(three rows and three columns or six expansions in all) by Laplace's rule 
and comparing each result with the definition of a determinant. For 
a proof of Laplace's expansion for nth-order determinants, see Ref. 11 
at the end of the text. 

24. Theorems Regarding the Expansion of Determinants. By 
means of the definition of a determinant and the Laplacian expansion, 
the following theorems are easily proved. In work with numerical 
determinants, the value of the following theorems cannot be over- 
emphasized. 

(a) A determinant is changed in sign by the interchange of any 
two of its columns (or rows). For example, 



0u 012 013 014 

021 ^22 023 024 

031 032 033 034 

041 042 043 044 



011 013 012 014 

021 023 022 024 

031 033 032 034 

041 043 042 044 



(b) A determinant is zero if any two of its rows or any two of its 
columns are alike. For example, 



= 0. 



(c) A determinant can be expanded (in Laplace's expansion) by the 
elements of any row or any column. 

(d) The value of a determinant is not altered if the rows be written 
as columns, and the columns as rows. For example, 



3441 
2122 
0354 
9176 


= 


3209 
4131 
4257 
1246 



(e) A common factor of all the elements of any row or column of a 
determinant may be divided out of the elements and placed as a factor 
before the new determinant. For example, 



1322 
2864 
9011 
7220 


r\ 


1322 
1432 
9011 
7220 



THEOREMS REGARDING EXPANSION OF DETERMINANTS 61 



(/) A determinant is not changed in value if we add to the elements 
of any row (column) the products of the corresponding elements of 
another row (column) by the same number. For example, multiplying 
the third column by 4 and adding to the second column 



2153 
3402 
1409 
1386 



2 1+4X5 5 3 

34+0 02 

14+0 09 

1 3+4X8 8 6 



2 21 5 3 
3402 
1409 
1 35 8 6 



By means of this theorem it is frequently possible to transform a 
determinant to an equivalent one in which all elements but one of a 
row (column) are zero, and thus simplify the expansion by Laplace's 
rule, as in example 1 below. The following examples further illustrate 
these theorems. 



EXAMPLE 1. Evaluate the determinant 

1111 
1234 
1 3 6 10 
1 4 10 20 



A = 



If the first column is multiplied by 1 and the result added succes- 
sively to the second, third, and fourth columns of the determinant, we 
have by theorem (/) 

100 

1123 

1259 

1 3 9 19 



A = 



Applying Laplace's rule to the first row we have 

1 2 3 



A = 



5 
9 



9 
19 



If the first column is first multiplied by 2 and added to the second 
and then multiplied by 3 and added to the third column, the deter- 
minant is 

10 



A = 



3 
10 



3 
10 



= 1. 



62 



DETERMINANTS 



EXAMPLE 2. Evaluate the determinant 



By theorem (e) 



By theorem (/), subtracting the first column from the second and third 
columns, 





7 3 


2 




= 


14 9 


4 


. 




21 27 


8 




7.3.2 


1 


1 1 




2 


3 2 




3 


9 4 



7.3.2 



1 

2 1 

3 6 1 



= 42. 



EXAMPLE 3. Evaluate the determinant 



A = 



7 7 
14 3 



3 21 1 



By theorems (e) and (6) 



A = 7 



EXAMPLE 4. Prove that 



1 1 7 
223 
3 3 1 



= 0. 



A = 



1 

y 



i 



= Ox -*)(*- *) (* - y). 



The factor theorem of elementary algebra states that, if OQX" + a\x n ~ l 
+ . . . + On vanishes for x = a, then x a is a factor of aox n + a\x n ~ l 
+ . . . + a n . The determinant A is a polynomial in x. (Also in y 
and z.) By theorem (6), A = for x = y. Hence by the factor 
theorem x y is a factor of A. By the same reasoning, z x and 
y z are factors. To see that there can be no additional factors, 
it is easy to compare the terms of the expanded product with the 
expanded determinant written as a polynomial. 

25. Multiplication of Determinants. We multiply two third-order 
determinants. For brevity n in this case is 3, but the method applies 
for any n. 



APPLICATION OF DETERMINANTS 



63 









011 


012 


013 




6ll 


612 hi 














021 


022 


023 




621 


622 623 


=3 












031 


032 


033 




hi 


6 32 ha 








0n b 


ti 4- 012 


bn 4- 013 bzi 0ii 


^12 


4- 012 622 4- 


013 hi 011 


&13 4" 012 hi 


4- 013 &S3 


021 b 


11 4- 022 


bn 4- 023 bzi 021 


6l2 


4- 022 622 4- 


023 &32 6(21 


hi 4" 022 ^23 


4" 023 633 


03i b 


11 4- 032 


621 4" 033 bn 081 


6l2 


4" 032 ^22 4" 


033 bat 031 


hi 4~ 032 ^23 


4- 033 633 



Or, in general, denoting an element of the product by (aft)*/, for an 
nth-order determinant 

*-n 



Right-hand and left-hand multiplications are equivalent, that is 



26. Application of Determinants to Non-Homogeneous 10 Linear 
Equations. In 21, determinants were related to the solution of two 
equations in two unknowns. We state, but do not prove, Cramer's 
rule for solving a system of n equations. If in the equations 



4- ... 4- 



the determinant 



a n \xi + . . . + a nn x n = fen, 
A = an .... ain 5^ 0, 



a n l . dnn 

the equations have exactly one solution, namely, 



#1 = 



(122) 



ki fli2 . . . 01n 


,X 2 = 


011 &1013 01n 


, , X n = 


011 012 ... 01n-l&l 


kz 022 ... 02n 




021 &2 023 ... 02n 




021022 . . .02n-l&2 


k n a n 2 . . . 0nn 




0nl^nn3. . . 0nn 




a.a n ,...a Bn .,* B 



(123) 



10 A homogeneous equation is one all of whose terms are of the same degree in the 
variables. For example, if we consider linear equations, 5x 4~ y 4- 6 = is a 
homogeneous equation; but 5x 4- y + 62 = 4 is non-homogeneous because of the 
term on the right side. 



64 DETERMINANTS 

The determinants which are the numerators of the fractions giving the 
values of #1, #2, . . . x n will be hereafter denoted respectively by the 
letters D\, D%, . . . D n , the denominator by A. 

If the determinant A of the coefficients of the unknowns in Eqs. 
(122) is zero, the investigation of solutions is more complicated. In 
this case, it is advantageous to employ the notion of the rank of a 
determinant. If a determinant of the nth order is not zero, it is said 
to be of order n. If the determinant is zero and also every (r + 1)- 
rowed minor formed from it is zero while there is at least one 
r-rowed minor which is not zero, then the determinant is said to be of 
rank r. For example, if a determinant of nth order is zero, but not 
all its minors of order n 1 are zero; the determinant is of rank 
n 1. If all the minors of order n 1 are zero but there is at least 
one minor of order n 2 which is not zero the nth-ordered determinant 
is of rank n 2. By means of the idea of rank the facts regarding 
the solution of n non-homogeneous linear equations in n unknowns 
are stated and illustrated as follows. 

(a) If the determinant A of the coefficients of the unknowns in 
Eqs. (122) is not zero, there exists a unique solution which is given by 
Cramer's rule, i.e., Eqs. (123). 

(b) If A is of rank r < n and any of the determinants J9i, Z?2, D n 
of Eqs. (123) are of rank greater than r, there is no solution of the 
system of Eqs. (122). 

EXAMPLE. Discuss the solution of the system of equations: 

x + y + z = 1, 

2x + 4y + 2z = 4, 

x - 3y + z = 2. 

In this system, the rank of A is 2. (The first and third columns being 
equal, A = 0, but 2-rowed minors which are not zero can be formed 
from A). The rank of at least one of the three determinants D\, #2, #3 
is 3. Consequently, no solution of the system exists. 

(c) If A is of rank r < n and the rank of D\, D<z, . . . D n does not 
exceed r, then there exist infinitely many sets of solutions of the system 
(122). The method of obtaining these sets is as follows: 

Since A is of rank r, the equations and variables in the equations 
can be so arranged that the upper left-hand r-rowed minor of A will 
be of rank r. Consider the first r equations of the rearranged system. 
Assign arbitrary values (say x' r +i, x'r+2 . . . x' n ) to the last n rvari- 



APPLICATION TO HOMOGENEOUS LINEAR EQUATIONS 65 

ables of these r equations, and transpose the results to the right-hand 
side of the equations. The system of the first r equations then is 



(124) 



k r a r 



Eqs. (124) may be solved for xi, . . . x r by Cramer's rule. It is then 
true that the values obtained for xi, . . . x r along with the values 
x'r+i, - x'n will satisfy the remaining n r equations of the n 
equations in n unknowns. But ff' r +i, . . x' n are arbitrary, and con- 
sequently there exist infinitely many sets of solutions. 

EXAMPLE. Obtain sets of solutions of 

3* + 4y s 6w 1, 
4x + Sy - 2z Sw = 2, 
5* + 4y - z - lOw = 1, 
3x + Sy - 2z 6w = 2. 

It can be shown that A, DI, Dz, Z>3, and D are each of rank 2. Since 
3 4 



the minor M = 



4 8 



is of rank 2 (that is, M ^ 0), it is not neces- 



sary to rearrange either the equations or the unknowns. Assign the 
arbitrary values z\ and wi, respectively, to z and w, and write 

3x + 4y = 1 + zi + 6wi, 
4 X + Sy = 2 + 2zi 

Solving for x and y, we have 

x = 2wi, 

1 +21 



4 
By substitution in the last two equations of the given system, x = 

y z = 21, w = Wi is seen to be a solution of the system. 

' 4 

27. Application of Determinants to Homogeneous Linear Equa- 
tions. Consider the set of n homogeneous linear equations in n un- 
knowns 

#12*2 + . . . + 01n*n = 0, 

(125) 



0, J 



66 DETERMINANTS 

Eqs. (125) have the trivial solution x\ = #2 = . = x n = 0. A 
necessary and sufficient condition that (125) have a solution other than 
the trivial one x\ = #2 = . . . = x n = is that the determinant A of 
the coefficients vanish. Or in other words, if the system (125) has a 
solution other than x\ = #2 = . . = x n = then A = 0, and if A = 
then the system has a solution other than x\ = xz = = x n = 0. 

The non-trivial solutions of (125) are found in much the same way 
that the solutions of the non-homogeneous equations (122) were found 
under case (c) of 26. Suppose that A is of rank r < n. Then Eqs. 
(125) and the unknowns in the equations can be arranged so that an 
r-rowed minor of rank r appears in the upper left-hand corner of A. 
Consider the first r equations of the rearranged system. Assign 
arbitrary values (say #' r +i, . . . , x' n ) to the last n r variables of 
these r equations and transpose these terms to the right-hand side of 
the equations. We then have Eqs. (124) where ki = #2 = . . 
= k n = 0. These equations can be solved by Cramer's rule for 
xi t . . . , x r . It is then true that the values obtained for Xi, . . . , x r 
along with the arbitrary values #' r +i, . . . . , x f n will satisfy the 
remaining n r equations of the n equations in n unknowns. Since 
jc' r +i, . . . , x f n are arbitrary, there exist infinitely many sets of solu- 
tions. 

EXAMPLE. Obtain sets of solutions of 

3x + 4y z 6w = 0, 
4* + Sy - 2z - Sw = 0, 
5x + 4y z - IQw = 0, 
3x + 8y - 2z - 6w = 0, 

in which A is the same as in the previous example. Evidently A, 

3 4 

4 8 

rank 2, it is not necessary to rearrange either the equations or the 
unknowns in the equations. 

We write 

3x + 4y = z\ + 

4* + Sy = 22i 
Hence 

x = 2w it 



Z, and D are of rank 2. Since the minor M = 



is of 



OBTAINING THE PARTICULAR INTEGRAL 67 

z\ 

The values x = 2wi, y = --, z = zi,w = wi satisfy the system of equa- 
tions whose solution is desired. Since z\ and w\ are arbitrary, we have 
obtained sets of solutions. The case of m homogeneous or m non- 
homogeneous equations in n unknowns (m ^ n) is explained in Ref . 1 7 
at the end of the text. 

28. Application of Determinants in Obtaining the Particular Inte- 
gral or Steady-state Solution of Simultaneous Differential Equations 
with Constant Coefficients and Sinusoidal Applied Force. Let it be 
required to find the steady-state solution of the system of differential 
equations 

ii + . . . + zi n i n = E sin 

1*1 + + Z2 n i n = 0, 

(126) 



Znlil + . . . + 2 nn 4 = 0, 

where 



7 4 S*l 

z r8 i a = Ln,* + R r8 i 8 + I i4t 

dt CraJ Q 



and L r8 , R r8 , and C ra are constants. 

:t 
i a dt by 
p 



J Xt 4 

If -~ be denoted by pi a and / i a dt by -i 8 , then 
"I JQ 



= L r8 p 



The system of Eqs. (126) may represent either a linear electric cir- 
cuit network with a sinusoidal applied voltage in one mesh or a mechan- 
ical system of n degrees of freedom with a sinusoidal applied force 
somewhere in the system. 

The calculation of the particular integral or steady-state solution 
of such a system is reduced to algebraic computation by the following 
proof. By subtraction of Eqs. (26) and (27) of 11, it follows that 



sin wt = 



Let us first find the particular integral of the system of equations 



68 



DETERMINANTS 



Li 



(127) 



+ Z nn i n = 0. 

Since the result of F rs (p) operating on a function e** is a constant 
times * or 



= LrA* + R r , + * = constant X **, 



it is evident that i\, . . . , i n must be expressible as some linear com- 
bination of e 1 " 1 if the first equation of (127) is to be satifised for all 
values of time. Therefore, let us try as a solution 



i. = I*** (5 = 1,2,..., n). 



(128) 



For abbreviation let 



Substituting (128) in (127), we have 



p 

+ . . . + Xln/n = T"., 



X21/1 + . . + X 2n /n = 0, 



Xnl/1 + . . . + X nn / n = 0. 

Solving by determinants for any / (say /,), we have 



(129) 



where A(io>) is the determinant of the coefficients of the system (129) 
and Ai 9 (iu>) is the cofactor n of Xi, in A. If - be denoted by 



11 The cofactor 13 the ynippr t^kep with the algebraic sign determined by Laplace's 
expan$ipn, 



OBTAINING THE PARTICULAR INTEGRAL 69 



i. = .~ ,. (130) 



To obtain the solution of system (127) with r replaced by - ~, it 

2i 2i 

is only necessary to replace i by i in Eq. (130). Then the particular 
integral of (126) for /, is the difference of (130) and (130) with i replaced 
by i, or 



E \ 
*' = 2il 



Since A(ico) and A\t(iv>) are both polynomials in iw, both are com- 
plex numbers. Hence Zi,(io>) is a complex number (say a + bi). The 
complex number a + bi can be written re** where r is its modulus and 
its argument. But since Zi,( i<o) is obtained from Zi.(iw) by 
replacing i by i, it follows that Zi( iw) and Zi,(ico) are conjugate 
complex numbers. Thus if 

Zi,(io>) = re 10 , 
then 

Zi a (iw) = re~**. 

If these values for Zi a (ioo) and Zi,( ico) are substituted in Eq. (131) 



where | Zi a (ico) | is the absolute value of the complex number Zi,(+ tw) 
and is its argument. 

EXAMPLE. Compute the steady-state terms or particular inte- 
grals of the solution of differential equations (106-107). 

The symbols of Eqs. (129-132) for the Eqs. (106-107) then are 



f2W, ^4l2 = ^2, 0) = -, 

jL 
A(to>) AM 

-- 



70 DETERMINANTS 

Thus by Eq. (132) the steady-state terms, si, and 52,, of $1 and 52 are, 
respectively, 

sin (o>/ 



S2 ' = - 



sin (ut <f>) 



where $ = and Z = AfiM 2 o> 4 - (Mik 2 + M 2 ki + M 2 k 2 )u 2 + kik 2 . 
The values of SI B and $2*, of course, agree with those obtained in Eqs. 
(109) and (110). 

29. Application of Determinants in Obtaining the Complementary 
Function in the Solution of Simultaneous Differential Equations with 
Constant Coefficients. We now obtain the complementary functions 
for Eqs. (106-107) by what is frequently called the classical method. 
Since the coefficients of the differential equations are constants, we sub- 
stitute in (106-107) 

5i = Ci^', ] 

(133) 

52-CV.J 

Making the substitution (133) and dividing out the factor e mt , we have 

+ h+ k 2 )Ci - * 2 C"i = 0, 1 



(134) 
- k 2 Ci + (M 2 m 2 + k 2 )d = 0. 

These equations are of the form (125), and by 27 there exists a solu- 
tion of (134) (other than C\ = C'\ 0) only in case 

Mini 2 + ki + k 2 k 2 

, ,, 9 , , 

k 2 M 2 m 2 + k 2 



Eq. (135) is called the characteristic equation of the system of equa- 
tions (106-107). Evidently (135), which is Eq. (108a), has four roots. 
In 18 these roots have been found to be it coit, co 2 t. Since Eqs. 
(106-107) are linear, the complete complementary functions are 



5i = Lie + L 2 e M . , 



The eight arbitrary constants of Eqs. (136) are not independent. 
The relations between d and C'i, C 2 and C' 2 , 3 and C'a, etc., are 
given by either of Eqs. (134) when m has been replaced respectively 
by wit, wit, and W2t, W2t. 



OBTAINING THE COMPLEMENTARY FUNCTION 71 

From the first of (134) 

Ci = - 



+ ki + k* 
From the second of (134) 



. 

K2 

The last two relations are identical provided 

&2 M2C01 2 



But this equation is only the characteristic equation, with m = wii, 
written in different form, and hence is true. 2 is related to C'i in the 
same way that C\ is to C'i. The relation between 3 and C'a is either 



or 



The same relationship holds between Ct and C\. The imaginaries are 
eliminated from (136) by Eqs. (26) and (27). 

The method just illustrated is applicable to all systems of the form 
(126). The characteristic equation in this case is 

= 0, where X r . = L r .m + R ra + . 



nlX n 2 X n n 

However, if the system of differential equations is at all complicated, 
the calculation of C'i, Ci", C'", etc., in terms of d along with the 
evaluation of the arbitrary constants d(i = 1, 2, ... n) is so labori- 
ous that the classical method is practically useless, and recourse is 
had to the operational calculus of Chap. IV. 

Evidently the characteristic equation A(m) = 0, in a system of 
many degrees of freedom, is an algebraic equation of high degree in m. 
This equation plays an important r61e in the study of electrical and 
mechanical system. Tests for stability of systems, whose differential 
equations of motion are linear, are developed in terms of the character- 
istic equation. (See 47, problem 6.) The roots of the characteristic 
equation give at once the natural frequencies of vibration of the system 
being studied. For this reason, in applications of the operational 



72 



DETERMINANTS 



calculus, it is frequently necessary to obtain the roots of the character- 
istic equation. Sec. IV of this chapter is concerned with finding the 
roots of higher-degree algebraic equations. 



EXERCISES AND PROBLEMS 

1 . Some of the following systems of equations have one or more solutions. Obtain, 
by use of determinants, these solutions, 



(0) 



2x - 6y - z 
3* -f 4? 4- 2z 



0, 
0. 



(c) x + 2y + 3z + 2w = 34, 

-x + 3y -h 7z - 2w = 36, 

4* + Sy + 5z -f Sw = 65, 

-4* + 1y + 3z - Sw = 39. 

(e) * -f 2? + 3* + \w = 34, 
3* + 6y + 9z + \2w = 102, 
4x + 8? + 5z + 6w = 65, 
-4x -f 7y -f 3z + 4w = 39. 



(b) x + 2y + 3z + w = 34, 

-x -f 3? + 72 -f 2w = 36, 

4* + 8? -f 5z + 6w = 65, 

-4 + 7y + 32 + 4u> = 39. 

(d) -x - 2y + 3z + 4w = 0, 

# - 3y + 7z -f 6w = 0, 

4* - 8;y + 5z + 16w = 0, 

9x + y + 2 - 2w = 0. 

(/) * + 2y + 3z + 4w = 0, 

-* + 3? + 7z -f 2w = 0, 

4jc + 8? + 5z -f 6w = 0, 

-4x + 7y + 3z -f 4w = 0. 



2. Write the characteristic determinant for each of the systems of Eqs. (114) 
(115), and (116). 

3. Compute the steady-state solution for each of the systems of Eqs. (1 14), (115), 
and (116). 

4. In the case of a third-order determinant, Laplace's expansion is equivalent to 
adding the products indicated by the heavy lines and subtracting the products indi- 
cated by the dotted lines. That is 




#11(122033 + #21032013 + 031023012 
~ 012021033 ~ 011032023. 



Show that this graphical method fails for determinants of order higher than 3 unless 
certain relations hold between the elements. In a determinant of the fourth order, 
write down the relations that must hold between the elements in order that this 
may give the oarne result as Laplace's expansion, 



HARMONIC ANALYSIS 83 

(c) To determine the coefficient b' n (n = 1, 2, 3, . . . ) of any sine 
term, take the average of the k ordinates after having multiplied each 
by the sine of n times the angle at which it is taken. The k ordinates 
are either those whose abscissas are fc, * 2 > *3, . . . and 2?r or 0, ti 9 
*2> t k i. Care must be taken not to take the ordinates at both 
and 2?r. The reason for this becomes evident in the proof of these 
rules in 38. The coefficient b f n is twice this average. 

(d) To determine the coefficient a' n (n = 0, 1, 2, 3 . . .) of any 
cosine term take the average of the k ordinates after having multiplied 
each by the cosine of n times the angle at which it is taken. The value 
of a' n is twice this average. 

Case 2. If the curve or table of values does not repeat itself, con- 
sider only the section of the curve over which it is desired to make 
calculations. The steps to be followed are precisely (a), (b), (c), and 
(d) of case (1). 

Case 3. The Ordinates of oscillograms of steady-state alternating 
currents in most electrical machinery not only repeat themselves at 
intervals of 2?r, but also reverse in sign at intervals of TT. In other 
words, if the functional relation of such a curve is i = f(t) then 
f(t + TT) = f(t). The curve in Fig. 20 is an example. In the Fourier 
representation of such a curve only those harmonics (sines and co- 
sines) can occur which reverse in sign when t is increased by TT. Con- 
sequently, only odd harmonics are present. This observation reduces 
the number of calculations one-half. The procedure in this case is 
identical to that of case 1 except that the calculation is made for only 
one arch of the curve. The unit is so chosen that the length of this 
arch is ?r. The coefficients of all even harmonics are zero. 

Before proving the rules just given, we illustrate their application. 

EXAMPLE 1 . Let the function y = /(/) be given by the table of 
values: 

TT TT 3w Sw 3ir 7ir 

t= 0> i' 2' T' ^'7' T' T' 27r 

/(/) = 2, f, 1, \, 0, i 1, f, 2 

_97r5rllx 137T ?7T 157T 177T 

* ~ 4 ' 2 ' 4 ' ' 4 ' 2 ' 4 ' 4 ' et ' 

/(O = i 1, i 0, J, 1, |, 2, f.etc. 

Let us make an harmonic analysis of this function. 

Evidently from the table /(2ir - /) = /(<) Thus no sine terms 
can appear since sin n(2w t} ^ sin nt- Since cos n(2v t} = cos nt 



84 



FOURIER SERIES 



f or n = or a positive integer, all cosine terms may be present. The 
function is periodic, hence only one period of 2ir need be used. The 
calculations are best carried out in tabular form. (No confusion will 
result if the primes are omitted from a' n and b' n in computations.) 



/ 


y 


cos / 


y cos / 


cos 2t 


y cos 2t 


cos 3/ 


y cos 3/ 


jr 


3 


















0.707 


1.060 








-0.707 


-1.060 


4 


2 














Tf 


1 


0.000 


0.000 


~1 


-1 


0.000 


0.000 


2 
















3ir 


1 


















-0.707 


-0.354 








0.707 


0.354 


T 


2 














7T 





-1.000 


0.000 


1 





-1.000 


0.000 


5w 


1 


















-0.707 


-0.354 








0.707 


0.354 


T 


2 














3T 


















1 


0.000 


0.000 


-1 


-1 


0.000 


0.000 


T 
















TT 


3 


















0.707 


1.060 








-0.707 


-1.060 


T 


2 














27T 


2 


1.000 


2.000 


1 


2 


1.000 


2.000 




8 




3 412 









0.588 



















#o = 2 X average of the y column = 2. 
#1 = 2 X average of y cos t column = 0.853. 
02 = 2 X average of y cos 2/ column = 0. 
as = 2 X average of y cos 3/ column = 0.147. 

Thus y = /(O = 1 + 0.853 cos / + 0.147 cos 3* + The coor- 
dinates of every point (/, y) of the given function y = /(/) satisfy this 
equation approximately. 

EXAMPLE 2. In a later chapter (Vol. II, Chap. II) on the solution 
of non-linear differential equations, an harmonic analysis of an oscillo- 
gram of the current will need to be made. The series so obtained will 
be a check on the answer there obtained by the analytical solution 
of the non-linear circuit equation. Fig. 20 is the reproduction of such 
an oscillogram. By use of the method of 37 an harmonic analysis 
will be made for this graph. Evidently, by case 3, 37, no even 
harmonics are present. The computations in tabular form are as 
follows. The numerical values of i in the table are given in hun- 
dredths of an ampere. 



HARMONIC ANALYSIS 



85 



1 1 



^OOOOOOOOOOOOOOOOO 

1 1 1 1 III 1 



to 

c 



ooooooooo^oooooooo 

1 1 1 1 1 1 1 1 



1 1 1 



II 



\o o o o ^o o ^o 

"O O O O O O *O 
QOtoOtooOOOO 



^o o ^o o o o o 

O O O O O O ^O 
oOOOOtoOtOOO 



I 



f<5 

C 



o 

O 



O O O *H O O O O O *-i O O O O O 1-" O O 

I I I I I 



OOOOOOOOOOO-HCqcsCN-r-iOO 

1 1 1 1 1 1 1 1 



-<ooooooooooooooooo 

1 1 1 1 1 1 1 1 



OOOOOOOOOrHOOOOOOOO 



8 



OOOOO 



OOOOOOO 



.3 



S s 

g CN 

x 1 

CN ( 



CN CN 



00 
CN 

t". 

o 6 2 ^ 

CN I O 0\ 

CO O d 

II 

o" 



00 



00 ON 



I i 



8 8 



& & 



rt ** cd cd 
X X X X 



86 FOURIER SERIES 

Thus the Fourier expression for i is 
i = 3.240 sin 6 - 0.728 cos + 0.014 sin 30 + 0.977 cos 30 

- 0.245 sin 50 - 0.111 cos 50 + ... 
or, what is the same thing, 

i = 3.321 sin (0 - 1240') + 0.977 sin 3(0 + 2657') 

+ 0.269 sin 5(0 - 3128') + . . . 

Approximate values computed by means of this series are marked by 
the center of the small circles in Fig. 20. The higher harmonics are 
seen to be negligible from an engineering point of view. 

1 

.05 
.04 
.03 
.OS 
.01 




R-72.50hmi 

- 1*6 Ohms 

Variable inductive reactance 
E - 43 Volts 
w-877 Radians /$ec. 



40 



80 



120 




Chech points: 1. 25 .0075.7 

2. 70 .01827 

3. 90 .OJ965 

4. tto* .oun 

5. 150 .02061 



FIG. 20. Current Non-linear Circuit. 

38. Proof of Harmonic Analysis Rules. The procedure outlined 
in 37 for finding the Fourier coefficients from a set of points or from 
a curve is that of integrating graphically the expressions for the coeffi- 
cients. By Eq. (145) 



i r 2 ' 

= - / /(*') si 
"Jo 



sin nl'W. 



But a definite integral may be defined as 



lim V 

4 



where the range b a of the variable / is divided into k segments 



THEORY OF FOURIER SERIES 87 

b - a 

= (A/), and k is any value of / within or at the end points of the 

ith segment. From the above, we may write 
i' n = lim - "V /(iA*) sin (in A/) A/, 

Af-0 7T ~ ^J 

2 * 

= (approximately, for A. small) ^O /(*A/) sin (in&t) 

2w 4-4 

- <~1 

A/ 
(*Afl sin (inAfl ( = 1, 2, 3 . . .). 



The last formula written is the expression for b' n . We have thus 
provided proof for the first three rules of case 1, 37. Similarly. 



*"* 



f A/) cos (fAO ( = 1, 2, 3 . . .), 



'"* 



o'o =- 



The proof of the rule in case 3 is Ex. 1 at the end of Sec. Ill on 
Fourier series. 

From the definition of a definite integral, it is evident that the 
ordinates used in the numerical integrations may be taken at any 
point of the corresponding interval. However, it is simplest to take 
either the first points or the last points of all the intervals, that is, 
either 0, /i, fe f fo-i or h, /2, . . . /*-i, 2x. 

There are various ways of making harmonic analyses. The 
method of 3738 is as simple as any and at the same time displays 
clearly the theory of the process. Other methods are described in 
Ref. 22. If many analyses are to be made, machines called harmonic 
analyzers are employed. 

39. Theory of Fourier Series. Although attention has been 
focused primarily upon the expansion of engineering functions (defined 
in 30) in Fourier series, it is by no means implied that these functions 
are the most general ones which can be expanded in Fourier series. 
Much more general functions can be so represented. For example, 
the function /(/) to be expanded can, under proper restrictions, have 
an infinitude of discontinuities within a finite interval of /. The 
function may also have a finite number of points at which it becomes 



88 FOURIER SERIES 

infinite provided it becomes infinite in certain definite ways. The 
most general function which can be represented by a Fourier series is 
unknown. Consequently, at most, it is possible to state only sufficient 
conditions regarding the expansion of functions in Fourier series. 
Sufficient conditions, if they are not also necessary conditions, restrict 
the functions considered to a smaller class. 

The treatment of Fourier series thus far has been concerned with 
the formal expansion of engineering functions. It is the purpose of 
this section to state and illustrate some of the facts and theorems 
regarding (a) convergence, (b) differentiation and integration, and 
(c) manipulation of Fourier series. 

(a) Convergence. The definition of convergence of an infinite 
series of real continuous functions is readily recalled from the calculus. 
Let 



be an infinite series of real continuous functions. Let 
S n (f) = i(0 + u 2 (t) + . . . + u n (f) 



be the sum of the first n terms of the infinite series. If for some value 
of /(say ti) the sum S n (t\) approaches a limit S(t\) as n approaches 
infinity, then the infinite series in question is said to be convergent at 
t = /i. If a series is not convergent, it is said to be divergent. If the 
series converges for every value of / in the interval a ^ t ^ 6, the 
series is said to be convergent in the closed interval (a, b). The inter- 
val a < t < b is an open interval. 

The four following theorems regarding the convergence of Fourier 
series are important. 

1. The Fourier series representing the engineering function /(/) 
in the interval TT < t < TT converges to the value /(/) at any point / 
in the interval TT < / < ?r at which /(/) is continuous; it converges 
to the value 



at any point at which /(/) is discontinuous. At the end points of the 
interval ( ir, IT) the series converges to the value 



The symbols 
denote 



THEORY OF FOURIER SERIES 89 

To illustrate this theorem, we refer to 33, example 2. Let us extend 
the definition of the function of this example by defining 

/(-2r) -/(-T) =/(0) -/(T) = f(2*) = 0. 
By the above theorem the series 

4a ^?A 



2m - 1 



sin (2m - I)/ 



converges to the value of the function at every point / of the interval 

-27T g t g 2*. 

2. If /(/) is an engineering function in the interval TT < / < TT, the 
coefficients in the Fourier expansion of f(t) are less in absolute value 
(numerical value, regardless of sign) than C/n, where C is some posi- 
tive constant independent of n. To illustrate, in the expansion of the 
function in example 2 of 33 the coefficients 

4a 4a 4a 4a 

y *Jf -j > v/> - /,- ^ \ 

TT 3?r 5?r (2m I)TT 

are respectively less than 

C C C C C C 



I 9 2' 3' 4' 5" "'2m -I"*' 

where C is any number greater than 4a/?r such as Sa/V. 

3. If /(/) is a continuous engineering function in the interval 
-TT < / < TT and/(?r - 0) = /(-TT + 0) and if the derivative/'^) is an 
engineering function in the same interval, then the coefficients in the 
Fourier expansion of f(t) are less in absolute value than C/n 2 , where C 
is some positive constant independent of n. For example, consider 
the function 

=- 7r -/ for -7r<*^-^ 
= t for -l*$l 

f(f) =TT -t for ~ ^ ^ < 7T 

JL 

which satisfies the above conditions. Its Fourier expansion is 
-I sin t - - sin 3/ + ^ sin 5^ - sin It + . . .1 



90 FOURIER SERIES 

The absolute values of the coefficients 

1 o ^ o (1)4 

^> ^9 v > -9 > / <\9 * * * 

7T 3 2 7T 5 2 7T (2m 1) 2 7T 

are respectively less than 

* C C C C C C 



!2 2 2' 32* 42' 52' ' ' (2ro - I) 2 ' * ' ' 
where we may have 



4. If /(/) and its derivatives up to and including the (s l)th are 
continous engineering functions in the interval w < t < TT and 

/ w (-r + 0) = /<> - 0)[r = 0, 1, ... (5 - 1)], 

and if the sth derivative is an engineering function in TT < t < TT, then 
the coefficients in the Fourier expansion of f(f) are less in absolute value 
C 

than Tfj[, where C is some positive number independent of n. To 
n 

illustrate, let 

Then 

/"(O = 



Now/(/) and its first three derivatives are engineering functions, and 



Thus in this case s 1 = 2, or 5+1 =4. The expansion of /(/) ia 
found to be 

cos J - cos 2/ + cos 3/ - . . . . 
2 o 



SUMMARY 91 

The absolute values of , - , , . . . , are respectively less than 

c c c c 

may be 2 * 



(b) Differentiation and integration of Fourier series. Sufficient con- 
ditions for the legitimacy of differentiating and integrating Fourier 
series are the following. If an engineering function f(t) is continuous 
for all values of / in the interval TT g t ^ TT and if /( ir + 0) = 
/(TT 0) , then the first derivative of the function is equal to the deriva- 
tive of its Fourier expansion. 

If /(/) is an engineering function, then the integral of /(/) is equal 
to the integral of the Fourier expansion of /(/). 

(c) Manipulation of Fourier series. It was stated in 35 that the 
Fourier expansion of an engineering function /(/) in the interval 
TT < / < TT is unique. It can also be shown that the Fourier expan- 
sion of 34 for the interval < / < 2w is unique. The theorems of 
this section have been stated for the interval TT to TT. The Fourier 
development for this interval is frequently called the Fourier expansion, 
and all other Fourier expansions are viewed as special cases of this one. 
If the theorems of this section are desired for the interval to 2?r they 
are easily obtained by a linear change of independent variable r = t 
+TT. Then r = for t = TT and r = 2w for t = TT. Or we may 
leave the theorems as they are and change the function. That is, if 
information is desired regarding the convergence of f(t) in the interval 
(0, 2?r) let r = / TT and expand /(T) = f(t TT) by the formulas of 
33 for the interval TT < T < TT. Then apply the theorems as 
stated. 

In the derivation of formulas (142) and (143), it has been assumed 
that the series (141) possesses convergence of a nature (uniform con- 
vergence) that will permit the term by term integration there per- 
formed. (See Ref. 21 at the end of the text.) 

40. Summary. 

(a) To obtain the Fourier expansion of an engineering function 
defined in the interval ( -IT, IT) use Eqs. (142), (143), and (144). If the 
function is discontinuous (say at a point / = a) it is recalled from the 
calculus that 



(b) If the interval of expansion is (0, 2ir) or (0, IT) or ( /, I) or 
(0, 2)> then the coefficients are given respectively by Eqs. (145), or 
(146) and (147), or (148), or (149) and (150). 



92 FOURIER SERIES 

(c) Harmonic analyses may be performed by the method of 37. 

(d) Questions of convergence of Fourier expansions of engineering 
functions can be answered, in most cases, by the theorems of 39. 

EXERCISES AND PROBLEMS 

1. In case 3, 37, show that the formulas for a' n and b' n , when the interval is 
to T, are the formulas already derived for the interval to 2w. 

2. Obtain, for subsequent use, the Fourier expansions of the following functions. 



(a) /u 



Ans. f(t) = sin t sin 3t -f~ sin 5^ .... 
o o 



. /(/) = sin t - sin 2t -f \ sin 3t + ^ sin 
- | sin 6/ -f- 1 sin It + 



f(t) = cos / + - cos 3^ - cos 5/ + .... 

435 

to /(/) = e l , < / < 2ir. 

e ' = e!lj= - 1 ^ + FTT cos ' + ^ 



. .. 



(The derivative of ' is not equal to the derivative of the series. In fact, the derivative 
of the series does not converge.) 
(d) /(/) - / 2 . -* < t < TT. 

Ans. f(t) = Y - 4 cos / - cos 2/ + cos 3/ - . . . . 

Is the derivative of t* equal to the derivative of the series? 

3. Given that e int = cos nt + ism nt, show that the series (141) and formulas 
(145) are reducible to the forms: 

/(/) = ? a n e tnt . 



n- -oc 



SUMMARY 



93 



where 



2a n 



a n ib nt n > 
a_ n -f ib-n, n <0 



and thus 



f(t)e~ int dt, (n -0, 1, 2,...). 



This is the complex form for Fourier series. It will be used later in the text. 

4. Complete the problem described in 32. A synchronous machine is driven 
by a reciprocating engine, which supplies a torque that fluctuates periodically about 
an average value. It is desired to determine the angular position of the rotor of the 
synchronous machine as a function of time. The data are as follows: 

T, = 240,000 Ib-ft. per radian, 

Td = 1800 Ib-ft. per radian per sec., 

WR* 
/ _ - f w here WR 2 = 16,500 Ib-ft. 2 , 

O Li ,L 

co = 26.9 radians per sec. 

The engine torque may be found from the crank-effort curve, shown in Fig. 21. The 
crank effort is the tangential component of the total force transmitted to the crankpin 



5.0 



1.0 





















1.5 
1.0 
.5 
0' 
-.5 
-1.0 
-1.5 








\ 










j*~\_ 






/ 












/ \ 


\ 




/ 




\ 


Average < 


ffort line\ 


/ 


\ 


\ 


/ 






\ 


^_^ 




/ 




\ 


/ 








^ "V 


- -x 


_/ 






V-X 




K-3000, 


l"SOOOIb. 





























ISO 160 200 t40 

Crank angle - Mechanical degrees 
FIG. 21. Crank-effort Curve. 



by the connecting-rod. Thus, if the crank effort is denoted by e, and the radius of 
the crankpin by r, the engine torque /(/) is given by 



re. 



The crank effort as a function of crank angle is found from Fig. 21 by multiplying 
the ordinates y of the curve by a scale factor of 3000. These ordinates at 15 inter- 
vals are 





2.23 
180 
2.05 



15 
1.63 
195 
1.41 



30 
1.35 
210 
1.55 



45 
0.89 
225 
1.59 



60 
1.21 
240 
1.39 



75 
1.83 
255 
1.48 



90 
2.20 
270 
1.44 



105 
3.05 
285 
1.32 



120 
3.37 
300 
1.71 



135 
2.99 
315 
2.31 



150 
2.74 
330 
2.99 



165 
2.48 
345 
2.99 



94 SOLUTION OF HIGHER DEGREE EQUATIONS 

The radius of the crankpin is 1 ft. 

First show by a Fourier analysis that the engine torque is approximately 

/(O = 6000 + 930 sin w/ - 330 cos ut 

- 2310 sin 2o>/ + 462 cos 2ut 

- 720 sin 3w/ + 1110 cos 3o>J 

360 sin 4w/ + 30 cos 4w* -f . . . . 

The differential equation of motion is then 

d*0 de 

I h Td h T 9 B = Taw + T^ut + 6000 + 930 sin at - 300 cos ut - 

d/ 2 dt 

(For the exact differential equation see Vol. II, Chap. I.) We are not primarily 
interested in the complementary function because the transient disturbance it repre- 
sents soon dies away. Find the instantaneous displacement 6 given by the particular 
integral. The periodic portion of the displacement is due to the periodic portion of 
the load. What is the maximum of the periodic displacement? 

Show that if the flywheel moment of interia is improperly chosen, the system may 
resonate with one of the harmonics of the engine torque. 

5. Suppose that the curve of Fig. 20 represents a voltage. Design a linear circuit 
such that the current due to this voltage will have a third harmonic component twice 
as great as the fundamental. 

IV 
SOLUTION OF HIGHER DEGREE AND TRANSCENDENTAL EQUATIONS 

The purpose of Sec. IV is to present suitable methods, from an 
engineering standpoint, of solving numerical higher-degree algebraic 
and transcendental equations. The need for the solution of such equa- 
tions has been partially indicated in the first part of the present 
chapter. The characteristic equations of simultaneous differential 
equations are likely to be of at least the fourth degree. Moreover, in 
vibrating mechanical and oscillating electrical systems the roots of the 
corresponding characteristic equations are generally complex quanti- 
ties. Hence, it is frequently necessary to obtain all the complex roots 
of fourth-, sixth-, and higher-degree algebraic equations. Further 
need for the roots of such equations will arise in the study of Heaviside's 
operational calculus. 

We desire solutions without a burden of formulas. Newton's 
method is employed for the solution of transcendental equations and 
Graeffe's method for the solution of higher-degree algebraic equations. 
The real roots of both transcendental and algebraic equations can be 
found by the method of successive approximations. This method has 
the merit that it rests on very little theory. It has the disadvantage 
that it is long and tedious if the equation is complicated. 



NEWTON'S METHOD 



41. Nature of Solutions of Algebraic Equations. Before develop- 
ing these methods, the following facts should be noted : 

(a) Algebraic formulas exist for the solution of the general quad- 
ratic, cubic, and quartic equations with literal (letter) coefficients. 
(See Ref. 24 at the end of the text.) 

(b) No formulas exist for the solution of a general algebraic equa- 
tion with literal coefficients if it is of higher degree than the fourth. 

(c) Any rational integral equation aox n + aix n ~ 1 + . . . + a n -\x + n 
= 0, whose coefficients are real or complex numbers has exactly 
n roots. These roots may be either real or complex. 

(d) An equation aox n + aix n ~ l + . . . + a n _i# + a n = 0, where n is 
a positive odd integer and the coefficients in the equation are real, 
always has one real root. 

(e) The number of positive roots of aox n + a\x n ~ l + . . . + #n = 
(a's real) is either equal to the number of variations of signs of the a's 
or in less than this number of variations by an even integer (Descartes' 
rule of signs). 

42. Newton's Method. Newton's method of solving equations 
is of engineering value because it not only gives the real roots of alge- 
braic equations, but with equal ease it yields the real roots of tanscen- 
dental equations. Moreover, it is unnecessary to remember any 
formulas since those required are readily derived from the principles 
of the differential calculus. 

The development of New- 
ton's method is as follows. 
Suppose it is desired 
to find a solution of the 
equation f(x) = 0, when 
an approximate solution 
is known, as by estimate 
or from a rough graph. 
In Fig. 22 y = /(*) is 
shown and OS is the root 

sought. Let OB = xi be the approximate solution, A M is the tan- 
gent to y = f(x) at the point where x = x\. (M must be sufficiently 
near to 5 that 12 f'(x) j* for any value of x between 5 and B.) 

12 The derivative of a function with respect to its argument is denoted by a prime 
thus 

/'(*) - /(*) 




FIG. 22. Newton's Method. 



and f'(xi) is the value of the derivative when x 



96 SOLUTION OF HIGHER DEGREE EQUATIONS 

From the figure and the differential calculus 



AB , 



If B is near 



nn 
= OB 



is a closer approximation than #1 to the root S of f(x) = 0. If next 
X2 is used as an approximate root, in the same manner that xi has been 
employed, a third approximation to 5 is 



#3 = *2 



If this process is continued either the root of f(x) = 0, or a close 
approximation to it, is obtained. 

EXAMPLE 1. Find the real root of the transcendental equation 
/(#) = x \ sin x 1 = 0. 

Let us first find an approximate solution #1, by graphical means. 
Since this equation is the difference of two functions, it can be written 
in the form 

sin x = 2(x - 1). 

V 

The abscissa of the point of 
intersection of the graphs of 
the two equations 

y\ sin x 

x y 2 = 2(x - 1) 

is evidently a root of 

x ^ sin x 1 = 0. 

The values of ^i and y^ are 
plotted in Fig. 23, from which 
an approximate value x\ of 
p IG 23. the root is found to be 





SUCCESSIVE APPROXIMATIONS 97 

Then, applying Newton's method, 



and 

X3 , , 2 _ IM = LS _ U.5 -| sin (1.5 radians)-!] = 
/ (#2) 1 - | cos 1.5 

Since/ (1.498062) = 0.000616 the process need not be repeated again 
for #3 is a good approximation to the root. 

EXAMPLE 2. Solve the equation # 3 2x 1 = 0. Since 

/(I) =-2 
and 

=+3, 



a root must lie between x = 1 and x = 2. Take as an approximation 

xi = 1.6. 
Then 



Since 

/(1.618) =- 0.0002, 

the value #2 is a fair approximation to the root desired. Since 
x = 1.618 is an approximation root of x 3 2x 1, it follows from 
the factor theorem that x 1.618 is an approximate factor. Hence, 
upon division of /(#) by (x 1.618) it is found that 

y? - 2x - 1 = (x - 1.618)(* 2 + 1.618 * + 0.618). 

The other roots, which are roots of the quadratic factor, are obtained 
at once. 

43. Successive Approximations. The method of successive ap- 
proximations is based on the principle that a smooth curve, for a small 
interval of the independent variable, is almost a straight line within 
that interval. An example will make the method clear. The value 
x = 1.6 is an approximation to a root of 

x 3 - 2x - 1 = 0. 



98 



SOLUTION OF HIGHER DEGREE EQUATIONS 



Since 



the graph of 



/(1.6) =-0.104, 
/(1.7) = 0.513, 

y = x 3 - 2x - 1 



crosses the #-axis between x = 1.6 and x = 1.7. If the graph is 
assumed to be a straight line through the points (1.6, 0.104) and 
(1.7, 0.513), we have the relations shown in Fig. 24. From the figure 

x 2 0.104 



0.1 0.617' 



or 



x 2 = 0.0168. 

Hence a more accurate value of the solution is 
x = 1.6 + x 2 = 1.6168. 



(1.7 1 .513) 



By continuing the process, a closer approximation can be obtained. 

Newton's method and the 
method of successive approxima- 
tion yield directly only the real 
roots. But the complex roots 
are the important ones in ob- 
taining the periods of vibrations 
and oscillations in many elec- 
trical and mechanical problems. 
Since Graeffe's method gives all 
the roots of any algebraic equa- 
tion, it is of great value. 
44. Underlying Principle of Graeffe's Root-squaring Method. 

The underlying principle of Graeffe's method is readily explained 

by means of a quadratic equation whose roots are real and distinct. 

Consider the equation 

x 2 + 10.1* +1 = 0. 

There exists 




*( 1.6, -.104) 

FIG. 24. 



1.7 
(x and y unit dissimilar) 



Suppose its roots (*2 = 10, *i = -^) are unknown, 
a simple routine method whereby we can transform 

* 2 + 10.1*+ 1 = 

into an equation whose roots will be some high even power (say the 
256th) of the roots of ** + 10.1* + 1 = 0. (It is, of course, not 



PRELIMINARY EXAMPLES 99 

necessary to know the roots of the original equation to do this.) The 
roots of the derived equation are then easily found, as may be seen 
by the following example. Let the derived equation be 

x 2 a\x + 02 = 0, 

where a\ and #2 are known by the above-mentioned routine process 
which will be described later. Since the roots of the last equation 
are x\ 25Q and #2 256 , we have 

x 2 - aix + a 2 = (x - *i 266 )(* - * 2 256 ) 

0. 



Hence 

*2 256 + * 2 256 = ai (151) 

and 

56 = a 2 . (152) 



Let X2 designate the root with larger absolute value. Since | x% \ >\ x\ \ 
we can neglect #i 256 in Eq. (151) and solve for #2. (That is, in the 
solution of (151), ( yV) 256 * s surely negligible in comparison with 
( 10) 256 .) Since X2 is now known, xi is easily determined from (152). 

If the roots of the original equation are not widely separated, it 
may be necessary for the roots of the derived equation to be those of 
the original raised to a still higher power. 

45. Preliminary Examples. We lead up to the general theory 
of Graeffe's method by the explanation of simple examples illustrating 
the different kinds of roots. The general theory will then be little 
more than a generalization of the notation used in the examples. 
Finally, from the general theory, a set of rules for the application of 
Graeffe's method will be derived. 

EXAMPLE 1. Roots real and distinct. Let the roots of 

x 2 + aix + a 2 = (153) 



be Xi and X2, |x2|>|xij. These roots with their signs changed are 
called the Encke roots of (153). Denote the Encke roots by x\ and #2. 
(The roots of the equation are denoted by bold-face type.) From the 
well-known relations between the roots and coefficients of a quadratic 
equation, we have 

ai = - (xi + x 2 ) = xi + X2, 

a 2 = XiX 2 = Xix 2 . (154) 



100 SOLUTION OF HIGHER DEGREE EQUATIONS 

By means of these relations, it is evident that the equation whose 
Encke roots are the mth power of the roots of (153) is 

X 2 + (Xi m + X2 m )x + Xi m X2 m = 0, 

and also 

x 2 + an* + a 22 = 0, (155) 

where an and a 22 are given by the routine process which will now be 
described. 

Write (153) in the form 

x 2 + a 2 = a\x. 
Squaring and rearranging, we have 

** - (a! 2 - 2a 2 )* 2 + a 2 2 = 0. 
If 

x 2 = - y (156) 

then 

y 2 + (ai 2 - 2a 2 )y + a 2 2 = 0. (157) 

By (156) the roots of (157) are the negative of the squares of (153), or 
the Encke roots of (157) are the squares of the Encke roots of (153). 
Applying, to (157), the process applied to (153), we have 



- 2a 2 ) 2 - 2a 2 % 2 + a 2 4 = 0. (158) 

If ;y 2 =-z, Eq. (158) is 

z 2 + [(ai 2 - 2a 2 ) 2 - 2a 2 2 ]z + a 2 4 = 0. (159) 



The Encke roots of (159) are the squares of the Encke roots of (157) 
and hence the fourth power of the Encke roots of (153). Eqs. (153), 
(157), and (159) may be respectively written 

x 2 + a\x + #2 = 0, 



+ a24 = . (160) 

By inspection, the following rule for writing a quadratic equation 
(called the derived equation), whose Encke roots are the squares of 
the Encke roots of a given equation, is seen to be as follows. The 
coefficient of any power of x in the derived equation is equal to the 
sum of the : 



PRELIMINARY EXAMPLES 



101 



(a) Square of the coefficient of the corresponding power of x in 
the given equation plus 

(b) The negative of twice the product formed by every pair of 
coefficients of powers of x which are equidistant from the power of x 
whose coefficient is being written. 

If the squaring process of Eqs. (160) is continued, we obtain equa- 
tions whose roots are the eighth, sixteenth, . . . and eventually the 
mth power of the roots of (153). Eq. (155) and the values of an and 
a22 are thus eventually obtained. But from 



and 

(x\X2) m = 0,22, (161) 

x\ and X2 can be found as explained in 44. 

Let it be required to solve, by the root-squaring method, 

x 2 + 10.1*+ 1 - 0. 

The probability of error is diminished by a tabular arrangement of 
work. 

TABLE I 



Squares of coefficients 


x 2 + 10.1* + 
1 102.01 


1 
1 


Minus double products 


- 2 










Second-power roots 


x 2 + 100 Olx + 


1 


Square of coefficients 


1 10002.0001 


1 


Minus double products. . . . 


2 










Fourth-power roots 


x 2 + lOOOO.OOOlx -f 


1 









By Eqs. (161), if Xi 4 is neglected, we have 
* 2 4 = 10000.0001 
X2 =db 10.000,000,002,5- 
xi =0.099,999,997,5+. 

By checking in x 2 + 10.1* + 1 = 0, the algebraic signs of the roots 
are seen to be negative. 

From an engineering point of view, the root-squaring process, in 
the numerical example just solved, was carried needlessly far. Con- 
sequently, a criterion is necessary for the termination of the root- 



102 SOLUTION OF HIGHER DEGREE EQUATIONS 

squaring process. To obtain such a criterion, for an equation with 
real and distinct roots, apply once more to Eq. (155) the rules following 
Eqs. (160) and obtain 

x 2 + [(xi m + X2 m } 2 - 2xi m x<2 m ]x + xi 2m x 2 2m = 0. (162) 

The Encke roots of (162) are the 2wth power of the Encke roots of 
(153). The Encke root #2 of (153) was determined from (155) by the 
relation 

X2 m + xi m X2 m = an. 

Now, for m sufficiently large, 



in Eq. (162), is practically 

(xi m + x 2 m ) 2 or an 2 . 

Hence to compute #2 of (153) from (162), we may write, for m suffi- 
ciently large, 

(#i m + x 2 m ) 2 = X2 2m = an 2 , 
or 

X2 m = an. 

But this is the same result as obtained from (155), and thus nothing 
was gained by an additional squaring. 

Thus, it is evident that if the root-squaring process is carried far 
enough the coefficients of the next derived equation are practically 
the squares of the corresponding coefficients of the preceding equation. 
Accordingly, we have the RULE FOR THE TERMINATION OF THE ROOT- 
SQUARING PROCESS FOR DISTINCT REAL ROOTS: 

(a) Determine beforehand the accuracy desired in the roots. 

(b) Cease the process when the double-product term obtained in 
an additional squaring has no effect on the root to the accuracy 
desired. Let the roots of the numerical example be desired to two 
decimal places. It is evident from Table I that the double-product 
term, 2, does not have sufficient effect on the coefficient of x in the 
second derived equation, 

x 2 + 10000.0001* +1=0 

to change the second decimal figure. Hence the last squaring was 
unessential. 

EXAMPLE 2. Roots real and equal. Let the double root of 

# 2 + 2aix + ai 2 = (163) 



PRELIMINARY EXAMPLES 103 

be Xj. Denote the Encke double root by x\. A criterion, by the 
root-squaring method, for the existence of coincident roots of a quad- 
ratic equation is easily obtained as follows. The equation whose 
Encke roots are the mth power of the Encke roots of (163) is 

x 2 + 2ai m x + ai 2m = 0. 

By the rules following Eqs. (160), which hold for all quadratic equa- 
tions, the next derived equation is 

x 2 + (4ai 2m - 2ai 2m )x + a^ m = x 2 + 2a^ m x + ai 4m = 0. 

(When the roots were distinct, the process ultimately led to coefficients 
which were the squares of the coefficients of the preceding equation.) 
In this case, the coefficient of x in 

x 2 + 2ai 2m x + at 4 = 

is only half the square of the coefficient of x in the preceding equation. 
Hence, the criterion for the quadratic equation is: If the coefficient 
of #1, in successive equations formed by the root-squaring process, is 
only half the square of the coefficient of x in the preceding equation, 
then the original Eq. (163) has a double root. 

The solution for the case of coincident roots is carried out exactly 
as for distinct roots. A table similar to Table I is constructed by the 
rules following Eqs. (160). Let the equation whose Encke roots are 
the rath power of the Encke roots of (163) be 

x 2 + anx + #22 = 0. 
Then 

xi m + Xi m = 2xi m = an 
and 

Xl 2m = #22. 

EXAMPLE 3. Roots complex. Let the Encke roots of 

x 2 + aix + a 2 = (164) 

be re ie and re~ ie . We desire a criterion, by the root-squaring process, 
for the existence of a pair of complex roots. The equation whose 
Encke roots are the mth power of the Encke roots of (164) is 

(x + r m e imd )(x + r m e~ tmd ) = x 2 + 2r cos mO + r 2 *" = 0. (165) 
If the root-squaring process is applied to the last equation, we have 
pc 2 + (4r 2 * cos 2 mB - 2r 2m )x + r 4 " 1 - # 2 + 2r 2m cos 2mBx + r* m = 0. (166) 



104 



SOLUTION OF HIGHER DEGREE EQUATIONS 



The criterion for a pair of complex roots now consists of two observa- 
tions : 

(a) Since the angle 6 in (166) is doubled by each squaring of the 
roots, the coefficient of x in (166) will frequently fluctuate in sign if 
m is sufficiently large. 

(b) It is evident that in the coefficient, (4r 2m cos 2 mO 2r 2m ), of x 
in (166) the double-product term 2r 2m does not vanish in comparison 
with the squared coefficient 4r 2m cos 2 mO as m increases. Thus with 
complex roots the root-squaring process is not continued (as was the 
case with real roots) until the double-product term becomes negligible 
compared to the square of the corresponding coefficient of the pre- 
ceding equation. The point at which the process is stopped is made 
clear in the following example. 

Let the roots of 

x 2 - 2x + 2 = 



be found by Graeffe's method. 



TABLE II 



Second power. 



Fourth power. 



x* - 2x + 2 
1 4 4 
-4 



1 4 
1 16 
- 8 



1 



8 16 



The equation whose Encke roots are the mth power of the roots of 

x 2 - 2x + 2 = 



is 



x 2 + 2r m cos mB + r 2m x 2 + anx + 
If we stop with m = 2 in Table II, we have 

f 2m = f 4 = ^ = 4 



0. 



or 



Let the Encke roots of 



also be written 



r =db\/2. 
x 2 - 2x + 2 = 



GRAEFFE'S GENERAL THEORY 105 

From the relations between roots and coefficients of a quadratic 
equation 

u = 1. 
Since 



Thus the roots at this point are either db (1 zb i). By checking in 
in the original equation, the roots are seen to be 1 db i. 

If the process had been carried to m = 4, we have from Table II 

r 2m = r 8 = 0,22 = 16 
or 

r =db\/2. 

Thus the process was needlessly carried through the fourth power of 
the roots (in this case to illustrate the behavior of the coefficient of x). 

With the insight thus acquired into the behavior of Graeffe's 
method as applied to simple examples, we consider his method as 
applied to the equation of the nth degree. 

46. Graeffe's General Theory. By means of the introductory 
examples of 45, the illustrative examples of this section, and the 
rules stated in 47, any algebraic equation can be solved with very 
little recourse to the Graeffe's general theory. However, occasionally, 
questions arise which may not be taken care of by the rules. In this 
event, it is necessary to understand, and at times even to make slight 
extensions of, the general theory. 

Consider the equation 

x n + aix n ~ l + a 2 x n ' 2 + . . . + a n = 0. (167) 

This may be written 

x n + azx"- 2 + d4X n ~ 4 + ...= (a\x n ~ l + a^x n ~ 3 + as* 71 " 6 + ...) 
Squaring both sides and rearranging, we have 
X 2n _ ( ai 2 _ 2a 2 )* 2n - 2 + (a 2 2 - 2aia 3 + 2a 4 )* 2 *- 4 

- (a 3 2 - 2a 2 a 4 + 2aia 5 - 2a 6 )^ 2n ~ 6 + . . . = 0. (168) 

Let the roots of (167), arranged in the order of their ascending abso- 
lute values, be 

*1 *2t *3 - x n- 

These numbers with their signs changed are the Encke roots of (167). 
Denote them by 



106 



SOLUTION OF HIGHER DEGREE EQUATIONS 



If x 2 = y, Eq. (168) may be written 



t,n-2 



a 3 



+ 20105 
- 2a 6 . 



;y-3 + . . . = 0. 



(169) 



The Encke roots of (169) are the squares of the Encke roots of (167). 

By inspection of Eqs. (167-169), the law of formation of the coeffi- 
cients of (169) may be stated thus: The coefficient of . any power of y 
in (169) is found by adding to the square of the corresponding coeffi- 
cient in (167) twice the product of every pair of coefficients in (167) 
which are equally distant from the term considered, these products 
being taken with alternately negative and positive signs. An absent 
power of x is taken with the coefficient zero. 

This process may be repeated indefinitely, the resulting equations 
having Encke roots which are respectively the square, fourth power, 
eighth power, etc., of the Encke roots of (167). This process will 
now be applied to the solution of equations. 

Case L All roots real and distinct. Let the Encke roots of (167) 
be xi, X2, #3, (The larger the subscript, the larger is the absolute 
value of the Encke root.) The well-known relations between the 
coefficients and the roots of an wth-degree equation are: 



02 = X n X n -l + XnX n -2 + . . . + XlX 2 = [x n X n -i] 

03 = X n X n -lX n -2 + X n X n -lX n -3 + . . . + #1*2*3 



The notation on the right is defined by these equations. 

Making use of the last relations, we see that the equation whose 
Encke roots are the mth powers of those of (167) is 



Now since 



yfl 1 f<y JH I jyTl I 1 f/y. Wl/y. ., JTt 1 A>1 2 1 f\ 

"' i L"'t* J"* i^ L w "*ii 1 \" i^ ~~ v/ 

\X n \> \X n -l | > |# n _2 |. , 



(170) 



it follows that x n m is enormously greater than x n -i m or # n -2 m for m 
sufficiently large. Thus the sum [# n m ] is very nearly equal to its first 
term # n m . In other words, 



[*"] = *n m (l + 



(171) 



GRAEFFE'S GENERAL THEORY 107 



where di is very small. Whence 



log | x n I = - log [x n *] - - log (1 + di). (172) 

m m 



Now 



are known quantities obtained by the root-squaring process or from 
repeated applications of the rule following Eq. (169). If log (1 + d\) 
is neglected in (172), the value of | x n \ is obtained at once. Next, 

[* n m *n-i m ] = *n m *n-i m (l + <fe), (173) 

where d% is very small. This equation is written 

log | Xn-l | = - log [x^Xn-i*] - ~ log [* n >] -- log (1 + J 2 ). (174) 

m mm 

If log (1 + ^2) is neglected, the value of | x n -i \ is obtained. This 
process is continued until the absolute values of all the Encke roots 
are found. To obtain the roots of (167) it is necessary only to attach 
the proper algebraic sign to the absolute values determined by (171), 
(173) .... Whether the sign of x n is positive or negative can be 
determined by the substitution of x w in the original equation. 

It is, of course, necessary to know when to cease increasing m, 
The time to stop doubling m is when another doubling gives an equa- 
tion whose roots are identical (to the number of figures originally 
decided upon) to those of the preceding equation. The criterion for 
stopping the root-squaring process may be stated thus: When the 
process yields coefficients in the next equation which are the squares 
of those of the preceding equation, to the accuracy required, the proc- 
ess is stopped. This criterion is established as follows. Suppose the 
root-squaring process is applied to the equation whose Encke roots 
are already the wth power of the roots of the original equation. We 
then obtain, corresponding to Eq. (171), the relation 

[x n 2m ] = *n 2m (l + i). (i small.) 

But by the definition of [x n 2m ] it is evident that, for 2m sufficiently 
large, 

[x n 2m ] = [*n m ] 2 (l + fe). (2 small.) 

Replacing in this equation [x n 2m ] by its equivalent value from the next 
to the last equation, we have 



108 SOLUTION OF HIGHER DEGREE EQUATIONS 

If di and 62 are negligible, 

But this is the value obtained from Eq. (171). Consequently, nothing 
is gained by an additional squaring of the roots when m has reached c 
value such that 

[*n 2m ] = [*n1 2 (l + 82) (82 small), 

that is, when the process yields for the coefficient of x n ~ l in the next 
equation a value which is the square of the corresponding coefficiem 
in the preceding equation, to the accuracy required. The reasoning 
is identical for the remaining Encke roots # n -i, . . . , #1. 

EXAMPLE 1. Solve 

x 4 + 1.18* 3 - 1.856* 2 - 0.396* + 0.072 = 0. 

The following tabulation, in view of the rule following Eq. (169), is 
self-explanatory. 

TABLE III 





X* 


X 3 


* 2 


X 


x 


1st 


1 


1.18 


-1.856 


-0.396 


0.072 
















1 


1.3924 
3.712 


3.444736 
0.93456 
0.144 


0.156816 
0.267264 


0.5184 X 1C- 2 


2nd 


1 


5.1044 


4.523230 


0.42408 


0.5184 X lO- 2 




1 


26.0549 
-9.0466 


20.4602 
-4.3293 
0.0104 


0.179844 
-0.046898 


0. 268739 X 10 - 


4th 


1 


17.0083 


16.1413 


0.132946 


0.268739 X 10 ~ 
















1 


289.2823 
-32.2826 


260.542 
-4.522 
0.000 


1.76746 X lO- 2 
-0.08676 X lO- 2 


7. 22206X10-" 


8th ... 


1 


256.9997 


256.020 


1.6807 X 10- 2 


7.22206 X 10~ 1C 
















1 


66,048.8 
-512.0 


65,546.2 
-8.6 


2.8248 X 10 -* 
-0.0037 X 10~ 4 


52.1582 X lO- 20 


16th.... 


1 


65,536.8 


65,537.6 


2.8211 X 10-* 


52.1582 X 10 - 20 



GRAEFFE'S GENERAL THEORY 109 

By Eqs. (172), (174), etc., 

log I x n I = - log [x m ] = log 65536.8 = (4.81649) = 0.30103. 
m lo 16 

x n = 2.000. 

log | * n -i I = ~ (log^n^n-i*] - log[*n m D = ^(4.81649 - 4.81649) 
m lo 

= 0.0000. 
x n -i= 1.000. 

log I # n -2 I = (log [# n m # n -l m #n-2 m ] log [# n m # n _i m ]) 
Wl 

= i^ (6,45042 - 10 - 4.81649) = 9.47712 - 10. 

X n _ 2 = 0.3000. 
log I # n _3 | = (log [# n m *n-l m *n-2 m #n-3 m ] ~ log [# n m # n -.l m # n _ 2 m ]) 

m 



1.71732 - 20 - (6.45042 - 10)] = 9.07918 - 10. 
# n _3 =d= 0.1200. 

Substituting in the original equation, the roots are ( 2, 1, 0.3, 
0.12). 

If Barlow's Table of Squares, Cubes, etc., and Crelle's Multiplication 
Tables are available, the root-squaring tables are rapidly constructed 
and no questions of accuracy arise. If no tables or computing 
machines are available, the slide rule is, in general, sufficiently accu- 
rate for the construction of the root-squaring tables. This is due to 
the fact that the roots are largely determined by the exponent of 10 
in the values of the coefficients of the wth-power equation. However, 
in the application of formulas (171), (173), . . . , logarithmic tables 
should be used which have one more place or decimal figure than the 
decimal place accuracy desired in the root. A summary of the pro- 
cedure for all cases is given in 47. 

Case 2. All roots real, two or more being equal. For definiteness, 
let the Encke roots arranged in descending order of absolute magni- 
tude be # n , #n-i, #n-2, . . . *i, with x n ~i = # n _2. The equation cor- 
responding to (170) is 

* n + [*n m ]* w ~ 1 + ^^ . . = 0. (175) 

This equation, for m sufficiently large, is approximately 

. . . 0. (176) 



110 SOLUTION OF HIGHER DEGREE EQUATIONS 

Eq. (176) is called the dominant of (175) since only those terms in 
each coefficient are retained which greatly exceed, for m sufficiently 
large, the sum of all the others. (The dominant terms of [#n m #n-i m ] 
are x n m x n -i m + * n w *n-2 m = 2# n m *-i m .) 

In the next equation of the root-squaring process, the coefficient of 
#n-2 w ;u b e 

O r 2m r ,2m 
LX n X n l , 

which is only half the square of the corresponding coefficient. (When 
the roots were distinct, the process ultimately led to coefficients which 
were the squares of the preceding ones.) Hence the rule: If the 
coefficient of x n ~ s in successive equations is only half the expected 
value, then the 5th and (s + l)th roots are equal. 

From the formation of the coefficients of (176), the solution for the 
repeated root is apparent. Evidently the 2mth power of the repeated 
root is the quotient of the coefficients on cither side of the irregular 
coefficient. (The coefficient of x n ~ 3 is in the numerator.) 

If two of the roots of an equation are equal in absolute value but 
opposite in sign, it is evident, from Eq. (172), that we still have case 2. 

If three of the Encke roots are equal (say # n _i = jc n _2 = # n _s), 
then the dominant equation is 

x n + x n m x n ~ l + 3x n m x n -i m x n -* + 3x n m x n -i 2m x n - 3 + . . . = 0. 

In the next derived equation the coefficient of x n ~ 2 will be 3jc /1 2m x n _ 1 2m 
which is only one-third the expected value if all roots were distinct. 
Thus # n -i may be computed by forming the quotient of the coefficients 
immediately following and immediately preceding the coefficient of 
*~ 2 . 

EXAMPLE 2. Solve 

x* - 3.80* 3 + 3.17* 2 + 0.92 x - 0.12 = 0. 
Table IV is constructed by the rule following Eq. (169). 

The next doubling of m would merely square each coefficient to 
four-place accuracy, except that of x 3 . Now 512 = J-32 2 , whence 
the two largest roots are equal. These are 

x n = * n -i = (6.553 -10 4 /1.000) M = 2.000. 
#2 = (4.304/6.553 -10 4 ) H = 0.3000. 
*i = (4.300 -10- 8 /4.304) H = 0.1000. 

There remains some question as to sign. By trial, it is found that the 
roots are 2, 2, 0.3, and 0.1. 



GRAEFFE'S GENERAL THEORY 
TABLE IV 



ill 





* 4 


X* 


X* 


X 


X* 


1st 


1 


-3.80 


3.17 


92 


-0.12 
















1 


14.44 
-6.34 


10.049 
6.99 
-0.24 


. 8464 
. 7608 


0.0144 


2nd 


1 


8 10 


16 80 


1.607 


0.0144 
















1 


65.61 
-33.60 


282.24 
-26.03 
0.029 


2.58 
-0.48 


2.074 X 10 - 


4th 


1 


32 01 


256.24 


2.10 


2.074 X 10~* 
















1 


1024.6 

-512.5 


65,659.0 
-134.44 
0.00 


4.41 
-0.106 


4.3 X 10~ 8 


8th 


1 


512.1 


65,525.0 


4.304 


4.3 X 10~ 8 















Case 3. One or more real roots and one pair of complex roots. For 
definiteness, let the Encke roots of (167), (n = 4) arranged in descend- 
ing order be #, re ie , re~ i9 , # n -3- The equation whose Encke roots are 
the wth power of these is 

(* + *n m )(* + r m O(* + r m e~ ime )(x + * n _ 3 m ) = 0. (177) 
Performing the indicated multiplications, (177) becomes 
x 4 + (x n m + *n-3 m + 2r m cos md)x 3 

+ [r' 2m + 2r m (x n m + # n -3 m ) cos md + x n m x n ,z m ]x 2 
+ [(x n m + # n -3 m )r 2m + 2r m # n m jt n _3 m cos mB]x + x^Xn-^r 2 " = 0. (178) 

When m is large, the last equation is approximately its dominant 
equation 

x 4 + x n m x* + 2x n m r m cos m6x 2 + x n m r 2m x + * n m x n _ 3 m r 2m = 0. (179) 

If the n Encke roots, in descending order (one pair complex), of an 
equation are 

X n , re i$ , X n -3, 



112 SOLUTION OF HIGHER DEGREE EQUATIONS 

then its dominant equation is 

x n + x n m x n ~ l + 2x n m r m cos w0 x n ~ 2 + x n m r 2m x n ~ 3 

+ x n m r 2m x n -3 m x n -* + . . . = 0. (180) 

As w is increased, the coefficient of x n ~ 2 fluctuates in sign owing to 
the presence of the factor cos mO. (The angle w0 is doubled at each 
application of the root-squaring process. See example 3, 45. Hence, 
if m is sufficiently large, md will not always lie in the first and fourth 
quadrants.) This fluctuation in sign indicates a pair of complex roots. 
Evidently the 2m power of the modulus r is the quotient of the co- 
efficients on either side of the irregular coefficient. After r and x n 
have been found, # n _3 can be found from the coefficient of # n ~ 4 , and 
the remainder of the real roots from the coefficients of the lower 
powers of x. 

To obtain the complex roots after their modulus has been obtained, 
proceed as follows. Let 

re i9 = u + iv, 

re~ ie = u iv. 
The sum of the Encke roots of the original equation then is 

dl = X n + 2U + X n -3 + #n-4 + . . . + #1, 

or 

= il ~ $(x n + X n - 3+ *n-4 + . . . + *l). 

The value of v is computed from 

v = A/r 2 u 2 . 
EXAMPLE 3. Solve the equation 

f(x) = x 3 - 2x - S = 0. 

Construct Table V by means of the rule following Eq. (169). 

The fluctuation of the sign of the coefficient of x exhibits the pres- 
ence of a pair of complex roots, the modulus of which is less than the 
real root. The real root is 

(3.5467 - 10 20 )* 4 = 2.0944. 
Now 

/(2.0) = - 1.0 

/(2.1) = 0.06 



GRAEFFE'S GENERAL THEORY 
TABLE V 



113 





X 3 


X 2 


X 


x 


1st 


1 





-2.0 


5 














1 


0.0 
4.0 


4.0 
0.0 


25.0 


2nd 


1 


4.0 


4.0 


25.0 




1 


16.0 
-8.0 


16.0 
-200.0 


625.0 


4th 


t 


8.0 


-184.0 


625.0 














1 


64.0 
368.0 


33,856.0 
-10,000.0 


390,625.0 


8th 


1 


4.32 X 10 2 


2.3856 X 10 4 


3.90625 X 10 s 














1 


1.86624 X 10 5 
-0.47712 X 10 5 


5.69109 X 10 8 
-3.37500 X 10 8 


1.52588 X 10 U 


16th. .. 


1 


1.38912 X 10 5 


2.31609 X 10 8 


1.52588 X 10 11 














1 


1.92965 X 10 10 
-0.04632 X 10 10 


5.36427 X 10 18 
-4.23926 X 10 16 


2.328309 X 10 22 


32nd 


1 


1.88333 X 10 10 


1.12501 X 10 16 


2.32831 X 10 22 




1 


3.54693 X 10 20 
-0.00025 X 10 20 


1.2656 X 10 32 
-8.769952 X 10 32 


5.4210 X 10** 


64th 


1 


3.54668 X 10 20 


-7.5044 X 10 32 


5.4210 X 10 4 * 



whence there is a real root between x = 2.0 and x = 2.1. The sign 
of 2.0945 is therefore positive. Again, 

r 2 = (5.42M0 44 /3.5467-10 20 ) H4 = 2.387, 

u = + |-2.095 = 1.0472, 
u 2 = 1.097, 
3,2 ^ r 2 _ U 2 = 2.387 - 1.097 = 1.290, 

v= 1.1357, 



114 SOLUTION OF HIGHER DEGREE EQUATIONS 

whence, finally, 

x = 2.0944, -1.0472 db i 1.1357. 
More accurate values are 

x = 2.09455, -1.04728 i 1.13594. 

Case 3 is summarized in 47. 

Case 4' Four distinct complex roots. An equation of the fourth 
degree all of whose roots are, in general, complex may occur as the 
characteristic or auxiliary equation associated with the differential 
equations of an electric circuit such as Fig. 10, or of an oscillatory 
mechanical system possessing two degrees of freedom, as in problem 1, 
20. 

Let the Encke roots of 



= 

arranged in descending order be r2e l<t> * and rie i(t>l . The equation whose 
Encke roots are the rath power of these is 

(x + r 2 m e im + 2 )(x + T2 m e~ im(h )(x + n m e im * l )(x + n^~ <m01 ) = 0, 
or 

x* + 2(r2 m cos w02 + r\ m cos w</>i)# 3 

+ (r2 2m + 4r2 m ri m cos m#2 cos m0i + r\ 2m )x 2 
+ 2(r2 m ri 2m cos w</>2 + r\ m r >m cos m^ijx + n 2m r2 2m ~ 0. 
The dominant of the last equation is 

f 2 2m = 0. (181) 



Let us examine the behavior of the coefficients of (181) when the 
root-squaring process is again applied. By the rule following Eq. 
(169), we have 



[4(r2 m cos w02) 2 l 
- 2r 2 2m J 



f2 4 



1 cos w02)(f2 m cos mfa) 



X 2 



+ f4(ri m r2 2m cos 
I - 2ri 2m r 2 4m 

+ (rir 2 ) 4 - = 0. 



GRAEFFE'S GENERAL THEORY 115 

In the coefficients of x 3 and x the double-product terms do not neces- 
sarily vanish, in comparison with the square terms, as m is increased. 
These coefficients eventually fluctuate in sign. The coefficients of x 4 , 
x 2 , and of are regular (i.e., the double-product terms vanish in com- 
parison with the squared terms), and the root-squaring process is 
stopped by the application of the rule following Eq. (174) to the 
regular coefficients. The coefficient of x 2 in Eq. (181) is r 2 2tn . After 
the value of r 2 2 is obtained, the constant term yields the value of ri 2 . 
To obtain the real and imaginary parts of the Encke roots, let 



r\e^** 1 = ui =t ivi. 
The original equation then is 
* 4 + 2(ui + u 2 )x* + (ui 2 + vi 2 + 4uiU 2 + u<2 2 + V 2 2 )x 2 

+ 2[w 2 (i 2 + *>i 2 ) + n,(u 2 2 + v 2 2 )]x + ( Ui 2 + Vi 2 ) (u 2 2 + v 2 2 ) = 0. (182) 
The relations between the Encke roots and coefficients give: 

ai, (183) 

fla, (184) 

ci2 r\ 2 r 2 2 . (185) 

Finally, the values of v\ and v 2 are given by the relations 



= db Vrr -- 



r, 



^2 = db V r2 2 ^2 2 . 

The algebraic signs of all the roots are determined by substitution in 
the original equation. 

EXAMPLE 4. Let us obtain, to one decimal place, the roots of 
x 4 + 25.5* 3 + 685x 2 + 8150* + 47,400 = 0. 

We construct Table VI by means of the rule following Eq. (169). 
Where the double-product terms may be neglected, an asterisk (*) 
appears. 

The fluctuation of the signs of the coefficients of x 3 and x exhibits 
the presence of two pairs of complex roots. From the dominant equa- 
tion and Table VI 

(r 2 2 ) 32 = 1.608 X 10 84 



116 SOLUTION OF HIGHER DEGREE EQUATIONS 

TABLE VI 





X* 


* 3 


X 2 


X 


K 


1st 


1 


2 55 X 10 


6.85 X 10 2 


8.15 X 10 3 


4.74 X 10 4 
















1 


6.5025 X 10 2 
-13.70 X 10 2 


4.6922 X 10* 
-4.156 X 10 6 
9.48 X 10 4 


6.6422 X 10 7 
-6.4938 X 10 7 


2.2468 X 10 9 


2nd 


1 


-7.198 X 10 2 


1.484 X 10 6 


1.485 X 10 6 


2.2468 X 10 9 
















1 


5.1811 X 10 6 
-2.968 X 10* 


2.2023 X 10 10 
2.136 X 10 
4.494 X 10* 


2.2023 X 10 12 
-6.6685 X 10 14 


5.0481 X 10 18 


4th 


1 


2.213 X 10 5 


2.8653 X 10 10 


-6.646 X 10 14 


5.0481 X 10 18 
















1 


4.897 X 10 10 
-5.731 X 10 10 


8.2099 X 10 20 
2.942 X 10 20 
1.010 X 10 la 


4.417 X 10 29 
-2.893 X 10 29 


2.548 X 10 37 


8th . . 


1 


-8.34 X 10 9 


1.1253 X 10' 1 


1 524 X 10 29 


2 548 X 10 37 


















6.956 X 10 19 
-2.251 X 10 21 


1.266 X 10 42 
2.542 X 10 39 
5.096 X 10 37 


2.323 X 10 68 
-5.735 X 10 58 


6.492 X 10 74 


16th.... 


1 


-2.181 X 10 21 


1.268 X 10 42 


-3.412 X 10 68 


6.492 X 10 74 




1 


4.757 X 10 42 
-2.536 X 10 42 


1.608 X 10 84 

* 

* 


1.164 X 10 117 
-1.646 X 10 117 


4.215 X 10 149 


32nd.... 


1 


2.221 X 10 42 


1.608 X 10 84 


-4.82 X 10 11 * 


4.215 X 10 149 



or 



or 



r 2 = 428.0 

(ri 2 r 2 2 ) 32 = 4.215 X 10 149 
n 2 = 110.74. 



Eqs. (183) and (184) give 



= 25.5 
= 8150, 



GRAEFFE'S GENERAL THEORY 11 7 

whence u\ = 8.39 

u 2 = 4.36 
and 

__ i A/ 12 2 i i /i oc 



v 2 = Vr 2 2 - 2 2 = 20.20. 
The roots are found to be 

-8.39 6.35i, -4.36 20.2i. 

Eq. (185) may be used as a check. 

Case 5. Six distinct complex roots. An equation with six distinct 
complex roots arises in the study of the motion of a rigid body, such 
as a top, which is in rotation but one point of which is fixed. The 
orientation of the body about the fixed point at any time / may be 
specified by three angles, for example, the Eulerian angles. 

The motion of the body following a displacement from an equi- 
librium configuration is described by three second-order differential 
equations, which are linear or frequently can be reduced to linear equa- 
tions by justifiable approximations (such as were employed in prob- 
lem 6, 10). Then the characteristic equation of the differential 
equation system is of the sixth degree and, in general, due to the oscil- 
latory character of the motion, has six distinct complex or imaginary 
roots. 

The study of transient oscillations in linear circuits having three 
branches, each of which has inductance, capacitance, and resistance, 
will in general necessitate the solution of a sixth-degree equation having 
complex roots. 

Although case 5 closely resembles case 4, yet because of its fre- 
quent occurrence a detailed study is made and an illustrative example 
solved. Let the Encke roots, arranged in descending order, of 

x 6 + aix* + a 2 x* + asx 3 + ax 2 + a<>x + a 6 = (186) 

be r^^\ r2e l< ^, and r\e^^. The equation whose Encke roots are 
the mih power of the Encke roots of (186) is 

(x + r 3 m e im **)(x + r 3 m e- im +')(x + r 2 m e im +*)(x + r 2 m e~ im +*) 

(x + nV* l )(* + n" 1 *" 1 "* 1 ) = 0, 
or 

x 6 + 2(ra m cos mfo + T2 m cos m<t>2 + r\ m cos m<t>i)x 5 
+ [(ra 2m + f2 2m + r i 2m ) + 4(r3 m f2 m cos mfc cos mfa 



118 SOLUTION OF HIGHER DEGREE EQUATIONS 

+ fs m ri m cos w</>3 cos m<i + r 2 m ri m cos w0 2 cos m<t>i)]x* 
+ 2[(r 3 m r2 2m + f3 m ri 2m ) cos w</> 3 + (r 2 m ri 2wi + r 2 m r 3 2m ) 
-f (ri m r 2 2m + r 1 m r 3 2m ) cos w<i + 4n m r 2 m r 3 m cos w< 3 cos w< 2 cos m<t>i]x* 
+ [(r 3 2m r2 2m + r 3 2m ri 2m + r 2 2m ri 2m ) + 4r 3 2m r 2 m n m cos w< 2 cos w<i 
+ 4r 2 27n r 3 m ri m cos w0i cos w</> 3 + 4n 2m r 3 tn ri m cos w< 2 cos w0 3 ]x 2 
+ 2(f 3 m ri 2m r 2 2m cos w< 3 + r 3 2m r2 m ri 2m cos ra< 2 + f3 2m ri m r 2 2m cos <t>i)x 
+ n 2m r 2 2m r 3 2m = 0. (187) 

The dominant equation of (187) is 

* 6 + (r 3 m cos w4> 3 )* 5 + r 3 2w x 4 + 2[(r 2 *Y 3 2m ) cosm^]^ 3 + r 3 2m r 2 2m jc 2 
+ 2[r 3 2m r 1 w r 2 2w cos m0i]jc + n 2m r 2 2w r 3 2m = 0. (188) 

When the root-squaring process has been applied once more by 
the application of the rule following Eq. (169), the following facts are 
in evidence. In the coefficients of x 5 , r 3 , and x the double-product 
terms do not necessarily vanish, in comparison with the squared terms, 
as w is increased. These coefficients fluctuate in sign for m sufficiently 
large. The coefficients of X Q , x 4 , x 2 , and x are regular and the root- 
squaring process is stopped by the application of the rule following 
Eq. (174) to the regular coefficients. 

The coefficient of x 4 in Eq. (188) is r 3 2m . The quotient of the 
coefficients of x 2 and x 4 gives r 2 2m , and finally the quotient of the co- 
efficients of x and x 2 is r\ 2m . 

To obtain the real and imaginary parts of the Encke roots, let 

r fc e ^ = u k iv kj (k = 1,2,3). 

The equation corresponding to (182) of case 4 is easily written down. 
After this equation has been written, a comparison of its coefficients 
with those of Eq. (186) gives the relations 



u 2 + u 3 ) = a lf (189) 

r 2 2 + 4w!W 2 + 4(^i + w 2 )w 3 + r 3 2 = a2 , (190) 

-f 4^ 2 + r 2 2 ) + (m + w 2 )r 3 2 ] = a 3 , 



= a 5 . (191) 

Since n 2 , r 2 2 , and r 3 2 are known, the simultaneous solution of (189), 
(191), and (say, 190) gives one or more values for lf w 2 , and 1*3. 



GRAEFFE'S GENERAL THEORY 119 

Double values for the variables may arise since one of the equations 
for the determination of the u's is quadratic. The two remaining 
equations of (189-191) may be used as a check to obtain the proper 
signs of the real parts of the roots and to reject the extraneous roots 
introduced by the quadratic equation. 

EXAMPLE 5. Let us solve the equation 
x* + 58* 5 + 1682* 4 + 11,64s* 3 + 42,436* 2 + 61,800* + 45,000 = 0. 

Table VII is constructed by the rule following Eq. (169). 
Comparison of the values of the coefficients of x 4 , x 2 , and #, as 
given in Table VII, with their expressions in (188) gives respectively 



( r3 2)i6 = 3,555 x 
(r 3 2r 2 2)i6 = 4.275 X 10 69 , 

(raWri 2 ) 16 = 2.829 X 10 74 , 
or 

r 3 2 = 1250.06, r 2 2 = 17.99, n 2 - 2.001. 

By Eqs. (189-191), 

3 = 25.00, U2 = 2.999, u\ - 1.002. 
From _ 

v k = Vr* 2 - u k \ (k = 1,2,3) 
we have 

z/3 = d= 25.00, v 2 = db 2.999, v\ = 1.002. 

The values found for the roots (not Encke roots) are 

x - - 25 db <25, -2.999 t'2.999, -1.002 dh il.002. 
The correct values are 



Case 6. Coincident complex roots. If two or more of the natural 
frequencies ( 12) of oscillation of the body described in case 5 co- 
incide, the characteristic equation will have coincident complex roots. 
The same statement is true for electrical circuits. Since resonance 
is so often either to be secured or avoided in engineering, but never 
ignored, this case also is important. 

Let the Encke roots, arranged in descending order, of 

x 5 + aix* + a,2X 3 + a& 2 + a& + a* = (192) 



120 



SOLUTION OF HIGHER DEGREE EQUATIONS 











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GRAEFFE'S GENERAL THEORY 121 

be xi and re^** (double roots). The equation whose Encke roots are 
the wth power of the Encke roots of (192) is 

(* + *i m )(* + r m e im +) 2 (x + r m e- im +) 2 = 0, 
or 

x 5 + (xi m + 4r m cos 



+ [4r w #i m cos w< + 2r 2m (2 + cos 
+ [4r 3w cos m<f> + 2r 2m (2 + cos 
+ (r 4m + 4r 3m #i m cos w</>)^ + r 4m ^i m = 0. (193) 

The dominant equation of (193) is 

x 5 + Xi m x 4 + 4r m #i m cos 
+ 2r 2m (2 + cos 2m<t>)xi m x 2 
+ f4jci m = 0. (194) 

(Coefficients in the dominant equation may consist of more than one 
term in case of multiple roots.) 

If the root-squaring process is applied once more to (194), it is 
found that: 

The coefficients of x 5 , x 4 , and x are regular while those of # 3 , 
x 2 , and x (adjacent terms) are irregular. 

The coefficients of x 3 and x fluctuate in sign for ra sufficiently 
large. The coefficient of x 2 eventually becomes positive as m is 
increased. In the 2wth equation, for m sufficiently large, the 
dominant part of the coefficient of x 2 is 

2xi 2m r 4m (2 + cos4w<). 

Its expected value, if regular, would be the square of the coefficient 
of x 2 in Eq. (194), that is 

4*i 2 -r 4 '*(2 + cos 2m<t>) 2 . 

The root-squaring process is stopped by the application of the 
rule following Eq. (174) to the regular coefficients. 

The value of x\ is obtained from the coefficient of x 4 , and in turn r 2 
is obtained from the constant term of (194). 

Next suppose r > | x\ |. Then the dominant equation of (193) is 

x 5 + (4r m cos w<)# 4 + 2r 2m (2 + cos 2w<)* 3 
+ (4r 3m cos m<t>)x 2 
+ r* m x + r 4m xi m = 0. (195) 



122 SOLUTION OF HIGHER DEGREE EQUATIONS 

In (195), the coefficients of x 5 , x, and x are regular while those of 
x 4 , x 3 , and x 2 are irregular. The values of r 2 and xi are given by the 
coefficients of the last two terms. 

EXAMPLE 6. Let us solve the equation 

x 5 + x 4 - 4x 3 - 16x 2 - 20* - 12 = 0. 

Table VIII is constructed by the rule following Eq. (169). 

Evidently in Table VIII the coefficients of r 5 , x*, and of are regular 
while those of # 3 , X 1 , and x are irregular. 

Comparison of the coefficients of x 4 and x in Table VIII and Eq. 
(194) gives 

Xl * 2 = 1.852 X 10 15 , 

^32( r 4)32 = 3 4225 X 10 34 , 
or 

xi = 2.9999 
and 

r 2 - 2.000. 

By the factor theorem, the root is 2.9999 = 3. The sum of the real 
parts of the Encke roots equals the coefficient of x 4 in the original 
equation. Consequently, if 

re **+ = u iv, 
then 

4u Xi 1 (#1 is the Encke root), 

4w = 1 + 2.9999 = 3.9999 = 4, 
= 1, 



The roots are 3, 1 =fc i, 1 i. 

Case 7. Seventh degree, two pairs of complex roots. As a final case, 
consider the seventh -degree equation 



x 7 + aix 6 + a 2 x 5 + azx* + a*x 3 + a 5 x 2 + a Q x + ai = 0, (196) 

whose Encke roots are #3,^*2, #1, rze****, and rie 14>l . The equation 
whose Encke roots are the mth power of the Encke roots of (196) is 

(x + # 3 m )(* + x 2 m )(x + xi m )(x + r 2 m e im +*)(x + r 2 m e~ imth ) times 

(x + ri m e im<f>l )(x + fi"**-** 1 ) = 0, 
or 
x 7 + a'ix 6 + a'< 2 x 5 + a' 3 * 4 + a' 4 x 3 + a' 5 x 2 + a'*x + a' 7 = O t (197) 



GRAEFFE'S GENERAL THEORY 



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124 SOLUTION OF HIGHER DEGREE EQUATIONS 

where 



cos w$2 cos w<i, 



cos m0i 
cos w</>2 



+ ri 2m + 



COS W02 



ri 



cos 



cos m$<i) 



cos m<f>i 



a 7 = 

Suppose first that 



Then the dominant equation of (197) is 



T2 > fl. 



COS W <f>2X 3 



Suppose secondly that 

r 2 > ri > | *3 | > | * 
Then the dominant equation of (197) is 

x 7 + 2f2 m cos w<2 x 6 + f2 2m x 5 



= 0. (198) 



) m = 0. (199) 



RULES FOR GRAEFFE'S METHOD 125 

Suppose finally that 

r2 > | #3 | > r\ > | x 2 | > | xi |. 
Then the dominant equation of (197) is 
x 7 + 2f2 m cos w02# 6 + r2 2m x 5 + r2 2m X3 m x* + 2r2 2m #3 m ri m cos mfax 3 

+ r2 2m X3 rn ri 2m x 2 + r2 2m X3 m X2 m ri 2m x + X3 m X2 m xi m r2 2m ri 2m = 0. (200) 

Inspection of Eqs. (198-200), and similar equations, leads to the fol- 
lowing conclusions: 

As the root-squaring process is continued, the number of coefficients 
which eventually fluctuate in sign is equal to the number of pairs of 
complex roots. The 2mth power of the modulus of any complex root 
is the quotient of the coefficients on either side of the irregular coeffi- 
cient. The mth power of one of the real roots is the quotient of two 
adjacent regular coefficients. Only the regular coefficients are used 
in the calculation of the moduli, and the root-squaring process is 
stopped by the application of the rule following Eq. (174) to the 
regular coefficients. 

The real and imaginary parts of the complex roots are determined 
from the moduli and the coefficients of the original equation in the 
following manner. Let the complex Encke roots of (196) be written 



u k ivk, (k = 1,2). 
Multiplying out the products 

(x + xz)(x + x 2 )(x + xi)(x + U L + ivi)(x+ui - ivi) 

(x + U2 + iv 2 )(x + u 2 - iv 2 ) = (201) 

and equating the coefficients of # 6 and x in (201) and (196) we have 
the relations: 

(x3 + #2 + #1) + 2(w 2 + u\) = m, 

+ wir 2 2 ) + r 2 2 ri 2 (x3X2 + #3*1 + #2*1) = fle- (202) 



Since U2 and u\ are the only unknowns in (202), these equations are 
sufficient to determine #2 and #1, except possibly for sign. The 
values of V2 and v\ are determined by 



v k = Vr/t 2 - w* 2 , (k = 1,2). 

47. Rules for Graeffe's Method. The three steps in the applica- 
tion of Graeffe's general theory to all algebraic equations are: (a) Con- 
struction of the table; (b) termination of the process of root-squaring; 
(c) calculation of the roots from the table. Steps (a) and (b) are the 



126 SOLUTION OF HIGHER DEGREE EQUATIONS 

same for all equations regardless of the nature of the roots, but step 
(c) depends upon the nature of the roots. 

(a) Construction of the table. Construct the numerical table by 
means of the law immediately following Eq. (169). As the successive 
equations are formed, any coefficient will behave in one of four ways. 
If m is sufficiently large, a coefficient in the next derived equation of 
the table may: 

(1) Be practically the square of its value in the preceding equa- 
tion. The coefficient is then said to be regular and the value 
approached is called its expected value. A coefficient is called 
irregular if it is not regular. 

(2) Approach l/q times its expected value. The degree of 
the equation is n and q = 2 or 3 or 4 or ... n. 

(3) Fluctuate in algebraic sign and at the same time not ap- 
proach its expected numerical value. 

(4) Fail to follow any obvious law of variation. 

(b) Termination of the process. Cease doubling m in the table 
when the regular coefficients are the squares of those of the preceding 
equation to the accuracy required and when the coefficients described 
in a(2) above, if any such are present, are 

- th (q = 2 or 3 or 4 . . . or n) 
9 

of the expected values to the accuracy required. 

(c) Calculation of the roots. Only the absolute value of a root is 
obtained from the table. The algebraic sign of a root is determined 
by substitution in the original equation. 

Case 1. All roots real and distinct. In this case, all coefficients of 
the table are regular. The roots are obtained from Eqs. (171), (173), 
etc., or, what is the same thing, the quotient of each power of x divided 
by the coefficient of the next higher power of x gives the mth power of 
a root. 

Case 2. All roots real, two or more coincident or equal in absolute 
value. If the coefficient of x n ~~ s in successive equations is only 

1/2 or - th , the square of the coefficient of x n ~ s in the preceding equa- 
tion, then the s and (s + 1) [or the s, 5 + 1, 5 + 2, . . . s + (q 1)] 
roots are equal in absolute value. The 2mth power of the multiple 

root of multiplicity q is : times the quotient of the coefficients 



RULES FOR GRAEFFE'S METHOD 127 

on either side of the irregular coefficient, i.e., the coefficient of x*~ J . 
(It is understood in this section that the coefficient of the lower power 
of x is in the numerator.) It is supposed now that the 1st, 2nd, ... 5, 
s + 1, . . . 5 + (q 1) roots have been found. Suppose the (s + q)th 
root is smaller than the multiple root. The mth power of the (s + q)th 
root is equal to the quotient of the coefficients of x n ~ ($+g) and x n ~ (s + q ~ l) . 
The remaining roots s + q + 1, s + q + 2, . . . n are, found as in 
case 1. (See Ex. 7, 47.) 

Case 3. One or more real roots and one pair of complex roots. The 
fluctuation in sign, for m sufficiently large, of one coefficient of the table 
indicates the presence of one pair of complex roots. The 2mth power 
of the modulus of these roots is equal to the quotient of the coefficients 
on either side of the irregular term. The real and imaginary parts of 
the complex root are determined by the equations following (180). 
The real roots are found from the remaining coefficients as in case 2. 

Cases 4 and & Four or six distinct complex roots. If all the roots 
are complex and distinct, the coefficients of odd powers of x fluctuate 
in sign. All other coefficients are regular. The modulus of any one 
of the roots is equal to the quotient of the coefficients on either side 
of the irregular term. The real parts of the roots are determined 
by means of Eqs. (183-185) or (189-191) and the imaginary parts by 
the relation 

v k = VV ~ "r (* = 1, 2 or 3). 

If all the roots of an nth-degree equation are complex and distinct, 
the procedure is the same. However, the relations between the real 
parts of the roots and the coefficients of the original equation must 
be determined. This can be done as indicated in Eq. (182). 

Case 6. One real and two coincident complex roots. An irregular 
coefficient, which apparently follows no law of variation and which 
stands between two coefficients fluctuating in sign, indicates one pair 
of coincident complex roots. The 4mth power of the modulus of the 
complex root is equal to the quotient of the coefficients on either side 
of the three irregular coefficients. The constant term of the equation 
is equal to r 4m *i m . The real and imaginary parts of the roots are 
determined by 

4u Xi = coefficient of x 4 , 



Case 7. General case. The number of coefficients which fluctuate 
in sign is equal to the number of pairs of distinct complex roots. The 



128 SOLUTION OP HIGHER DEGREE EQUATIONS 

2mth power of the modulus of any distinct complex root is the quotient 
of the coefficients on either side of the irregular coefficient. 

The mth power of one of the distinct real roots is the quotient of 
two adjacent regular coefficients. 

ty I / ty \ I 

The 2mth power of a real root of multiplicity q is equal to : 

? ! 

times the quotient of the coefficients on either side of the irregular 
coefficient satisfying the description in a(2). (Also see case 2.) 

Complex double roots are indicated by the presence of three adja- 
cent irregular coefficients, the outside two of which fluctuate in sign. 
The 4mth power of the modulus of the double roots is equal to the 
quotient of the coefficients on either side of the group of three irreg- 
ular coefficients. 

The real parts of the complex roots are obtained through the rela- 
tions between roots and coefficients of the original equation. (See 
Eqs. (201-202).) 

The imaginary parts of the complex roots are then found as in 
case 6. 

Any prescribed accuracy may be attained by Graeffe's method. If 
great accuracy is required, it may be advantageous to make a pre- 
liminary determination of the roots in order to see how many figures 
must be retained in squaring and multiplication processes to secure 
the accuracy required. If the equation is of high degree, it is advan- 
tageous to depress the original equation by removing all the real 
roots. High accuracy for the real roots may be obtained, with little 
labor, by the methods of 42-43. References by which the accuracy 
of complex roots can be improved are found at the end of the text. 
(See Ref. 25.) 

EXERCISES 

1. Solve the equations: 

(a) x 3 - 7* + 7 = 0. 
(6) *' - 3.5* + 1.6 = 0. 

(c) x 3 + * 2 - 2x - 1 = 0. 

(d) cos x(e x + c-*) + 2 = 0. 
(c) x* -f 4 sin x = 0. 

(/) cos x - 3* + 1 = 0. 

2. Solve the equations: 

(a) x* + 2* 3 - 12* 2 - 10* + 3 = 0. 

(b) ** - 16* 3 + 86* 2 - 176* + 105 = 0. 

(c) x* - 11.70* 8 -1- 5.96* 2 + 131.82* - 214.20 = 0. 

(d) *< + *- 5* 2 + 9* - 10 = 0. 



RULES FOR GRAEFFE'S METHOD 129 

3. Solve the equation : 

x 6 + x* - 33* 3 - 26jc 2 + 256* + 224 = 0. 

4. Solve the equation : 

x 7 - 6* 6 + 21** - 21* 3 + 32* 2 - 56* + 18 = 0. 

5. In the calculation of the transient field current of a synchronous machine by 
Heaviside's expansion theorem, it was necessary to find the roots of 

p* + 0.6193/> 4 + 0.1273/> 3 + 1.044 X I0~ 2 p* + 2.966 X_lO~ 4 p + 5.739 X 10~ 7 = 0. 

What are these roots? 

6. The roots of the characteristic equation of a simultaneous system of linear 
differential equations with constant coefficients characterize the motion (or currents) 
}f the physical system. If the real roots and real parts of the complex roots are 
legative or zero, the motion is limited in magnitude and is said to be stable. On 
:he other hand, if the roots have positive real parts, the motion is unstable. Fre- 
quently, in design, it is sufficient to know whether a system is stable. The criterion 
'or stability is much more easily applied than Graeffe's method. 

The characteristic equation of the differential equations defining the motion of a 
:ertain type of electric locomotive was found to be 

x 9 + 68.6* 6 + 785*< + 7213* 3 -f 50,700* 2 + 8200* + 435,000 = 0. 

[See Vol. II, Chap. I.) It is desired to determine the stability of the motion. 

The proof of the following criterion is found in Routh's Advanced Rigid Mechanics, 
3. 170. Criterion: Arrange the coefficients of 

P&" + pix*~* + P*x n -* + . . . + p n = 
in two rows 



PI Pa P& . . . 

Form a new row by cross-multiplication as follows: 
Pip 2 pops p\p\ pops 

PI ' PI ..... 

Form a fourth row by similar cross-multiplication on the second and third rows. 
The number of terms in successive rows decreases. Stop when one term is left. 

A necessary and sufficient condition that the equation have no roots whose real 
parts are positive in that all the terms in the first column have the same sign. The 
number of variations of sign in the first column is equal to the number of roots or 
pairs of complex roots with their real parts positive. 

Since only the signs are important, any row may be divided by a positive constant 
;o simplify multiplication, 



130 DIMENSIONAL ANALYSIS 

Applying the criterion to the locomotive equation we have: 

1 785 50,700 435,000 

68.6 7,213 8,200 

68.6 X 785 - 7,213 68.6 X 50,700 - 8,200 



6.86 721.3 820 

6.7986 505.S 4,350 

6.7986 X 721.3 - 6.86 X 505.8 6.7986 X 820 - 4,350 X 6.86 



6.7986 ' 6.7986 



6.7986 505.8 4,350 

2.1092 -35.692 

2.1092 -35.692 

6.2085 43.5 

6.2085 43.5 

-50.469 

-2195.4 

The change of sign occurring in column one indicates the presence of one pair of 
roots having positive real parts. The motion is unstable, but can be corrected by 
varying one of the parameters of the coefficients in the characteristic equation. The 
roots are found to be (approximately) 

x = - 56.7, -11.02, -0.92 db J9.07, -f- 0.47 J2.86. 
Is the motion stable when the characteristic equation is 

x* + 2.55 X 10x 3 + 6.85 X 10 2 * 2 + 8.15 X 10 3 x + 4.74 X 10* = 0? 
7. Prove the rule in case 2, 47. 

V 
DIMENSIONAL ANALYSIS 

The first four sections of the present chapter are seemingly unre- 
lated from a mathematical standpoint, but from an engineering point 
of view they are closely related since they form a single engineering 
problem-solving unit. To begin with, the mathematical formulation 
of the engineering problem may lead to one or more ordinary linear 
differential equations. Determinants afford systematic methods of 
manipulating simultaneous systems of differential and algebraic equa- 
tions. Complicated voltages and forces are expressible in Fourier 



USES AND NATURE OF DIMENSIONAL ANALYSIS 131 

series, and these series can be handled successfully in ordinary differ- 
ential equations. Finally, GraefiVs method gives a quick and accurate 
method of obtaining the roots of the characteristic equation. These 
roots then display the general nature of the motions of the system. 

A second problem-solving unit, which is independent of the first, 
is dimensional analysis. In certain respects, dimensional analysis is 
applicable to a wider range of problems than the methods so far 
explained. The principles and relations discussed up to this point are 
defined by systems of ordinary linear differential equations with con- 
stant coefficients. The problems solvable by dimensional analysis 
are subject to no such narrow restrictions. Much information regard- 
ing results of problems reducible to ordinary and partial linear and 
non-linear differential equations can be obtained by dimensional 
methods. In fact, these methods are applicabl^also to problems whose 
solutions do not depend upon differential equations. Moreover, the 
amount of labor and mathematical knowledge necessary for carrying 
out a dimensional analysis solution is usually small in comparison 
with that required for solutions otherwise obtained. 

The method, however, has the great disadvantage that its results, 
in general, give information much less complete than that obtained 
by the solution of differential equations. However, dimensional 
results are adequate in many engineering problems. 

48. Uses and Nature of Dismensional Analysis. The principal 
value of dimensional analysis lies in the following applications: 

(a) Checking equations. 

(b) Changing units. 

(c) Derivation of formulas. 

(d) Analysis of physical systems by use of models or physically 
similar systems. 

(e) Systematic experimentation. 

These topics are discussed in 50-56. 

Owing to the fact that a discussion of dimensional analysis in its 
complete generality takes on rather abstract aspects, the nature of 
such analysis is best perceived by the examination of results obtained 
by it in representative problems. But before writing down such results 
we first recall from elementary physics some ideas regarding dimen- 
sions. 

The dimensions of a physical quantity are concerned with its 
quality or kind and not with its magnitude. Underlying all dimen- 
sional analysis is the concept of the dimensional formula of a physical 
quantity, which is an expression showing the manner in which a group 



132 



DIMENSIONAL ANALYSIS 



of chosen fundamental units are combined to make a unit of that 
quantity. For example, in terms of length as a fundamental unit, 
the dimensions of area and volume are respectively (length) 2 and 
(length) 3 , or in symbols [A] = L 2 and [V] = L 3 . (This is read: 
" The dimensions of volume are length cubed.") If in any given 
problem involving n physical quantities (dimensions), r of them are 
expressed in terms of the (n r) remaining ones, the (n r) quanti- 
ties are called the fundamental dimensions or primary quantities, and 
the r quantities are called derived dimensions or secondary quantities. 
For example, in the problem of 6 the n quantities are acceleration, 
force, mass, length, velocity, and time. The fundamental quantities 
may be chosen to be length, mass, and time. Acceleration, force, and 
velocity are then derived quantities. Not every set of (n r) dimen- 
sions can be used as a fundamental system of units. The criterion 
for a fundamental system appears later, in 51. Length L, mass M, 
and time T, however, form a suitable system for mechanics. The 
dimensional formula for a quantity is derived from the definition of 
the quantity. 

In mechanics, we may have the following dimensional formulas: 

[Linear displacement x] = L, [Moment of inertia] = ML 2 

Linear velocity = - = LT~ l , [Angular momentum] = ML 2 T~\ 
L dt J T 

r rf2 n [Kinetic energy] =ML 2 T~ 2 , 

Linear acceleration =LT~ 2 , ,_ 
L dt 2 J [Force] =MLT~ 2 , 

[Linear momentum] =MLT~ l , [Work] =ML 2 T~ 2 , 



[Angle] = (dimensionless), 



[Power] = 

[Angular velocity] = T~ l , [Moment of force] = 

[Angular acceleration] = T~ 2 , [Density (p)] = 

In fluid mechanics, we may have the dimensional formulas: 

[Force 
Pressure gradient (G) = -^7- -f- Length | = ML- 2 T~ 2 , 



Area 



1 

lent = 
Kinematic viscosity (v) = ^ . = L 2 T~ l * 



[T-, 
Viscosity (p) = A -f- Velocity grad 



Area 



SOME REPRESENTATIVE RESULTS 133 

Suitable primary quantities in heat-flow problems are L, T, II 
(quantity of heat), and 6 (temperature), or L, M , T t and 0. Dimensional 
formulas for certain secondary quantities in terms of the former are; 



[Rate of heat transfer] = 

[Thermal capacity (per unit volume)] = 

[, ... (Rate of heat transfer per unit area)! 
Thermal conductivity - - - - 
1 emperature gradient J 

= HL- l T~ l e- 1 , 

or, in terms of the latter primary quantities, 

[Rate of heat transfer] = ML 2 T~ 3 , 
[Thermal capacity] = ML~ l T- 2 e~ l , 

[Thermal conductivity] = MLT- 3 6~ l . 

Suitable primary quantities in electrical problems are M, L, T 
and 8 (the dielectric constant). Dimensional formulas for certain 
secondary quantities are : 

[Charge] = &*L*M*T~ l , 

[Voltage] 

[Current] 

[Capacitance] = SL, 

[Resistance] = &- l L~ l T. 

The nature of dimensional analysis can now be seen by inspection 
of the results in representative problems. 

49. Some Representative Results. The methods of obtaining the 
results displayed in this section is the subject-matter of subsequent 
sections. 

1. Checking equations. The differential equations of problem 1, 
20, are (where primes indicate derivatives with respect to time) : 

M 2 s 2 " + k d (s' 2 - *'i) + k 2 (s 2 - si) = 0, 

" - k d (s' 2 - S\) + kiSi - k 2 (s 2 - Si) = 0, 



where Mi and M 2 are given in slugs, kd in pounds per unit velocity, and 
ki and k 2 in pounds per unit displacement. Let it be required to 
check dimensionally these equations. Owing to similarity of various 
terms, it is necessary to check only three. Since 

] = MLT~ 2 



134 DIMENSIONAL ANALYSIS 

X = MLT-* 



~\ r ~r nr* _ 2 

[k 2 s 2 ] = - - X L = MLT~ 2 

LJ 

the terms have the same dimensions and the equations are dimen- 
sionally correct. 

2. Changing units. Let it be required to convert Q kilowatts to its 
equivalent in feet, pounds, and seconds. By simple consideration of 
dimensions, the result can be shown to be 

^ , ., **** foot-pounds 

Q kilowatts = (27370 - ^ . 
seconds 

3. Derivation of formula: Oscillation of rotor. Eq. (140), 32, may 
be written 

70" + T d (9' - ) + T.(6 - /) = 



Suppose there are no torques on the rotor except the synchronizing 
and inertial torques, and let the synchronizing torque, T 8 (6 co/), 
be replaced by its more accurate value T 8 sin (6 o)/). Replacing 
(0 o)/) by a new variable 0, the equation becomes 

I<t>" + T 8 sin = 0. 

If the rotor is displaced from synchronous position and suddenly 
released, find its period of oscillation. The period t can be found by 
dimensional analysis to be 



where 0o is the amplitude of the periodic variation of 0. The quantity 
0o, since it is an angle, is dimensionless. The function F(0o), which is a 
function of a dimensionless argument, cannot be found by dimensional 
methods. Its behavior is easily determined experimentally, since for 
/ and T 8 fixed, / is a function of 0o alone. 

4. Derivation of formula: Amplitude of damped oscillation of a mass. 
Let the mass in Fig. 2, 10, be acted upon by the vertical force F 
sin a)t. Let the damping and spring constants be respectively kd 
and k. By dimensional analysis, it is easily shown that the amplitude 
of vibration A is 

A =?j 



SOME REPRESENTATIVE RESULTS 135 

The arguments, - - and ^, are dimensionless, and the function / 

* k 



cannot be determined by dimensional reasoning. The nature of / 
can be determined experimentally. 

In this case, the function / is easily found by other means (from 
the solution of the differential equation) to be 

ft 



L. ?**\ - 

r' *y~ 



In this example, much more complete information is easily obtained 
by the solution of the differential equation than by dimensional 
analysis. 

5. Derivation of formula: Propeller thrust. It is desired to deter- 
mine the thrust T of a screw propeller, which is so deeply immersed 
in water that there is no surface turbulence. The result for this 
problem is 



/Dn DS Dg\ 
\T' ~' S 2 /' 



where: D = diameter of the propeller, 
n = revolutions per minute, 
5 = speed of advance, 
g = acceleration of gravity, 
p = mass density of water, 
v = kinematic viscosity of water. 

The function /can be determined experimentally. 

6. Derivation of formula: Heat transfer. A solid body of certain 
shape is held at a constant temperature 6 and fixed in a stream of 
liquid which flows past it at a velocity V. It is required to find the 
rate of heat transfer R from the body to the liquid. If viscosity and 
surface conditions can be neglected, the result in this problem is 



R = k(M)lf \(-i- c ),r,r',r" 



,...], 



where k = thermal conductivity of the liquid, 

A0 = temperature difference, 
/ = a linear dimension (say the length), 
V = velocity of the stream, 
c = heat capacity of liquid per unit volume, 
r, r', r" . . . = ratios of dimensions of body (shape factors), and 
/ is an unknown function. 



136 DIMENSIONAL ANALYSIS 

The quantities /, c, k, and r, r' t r" . . . are constant for a given body 
and liquid. Thus the function / might be determined by measuring 

R T7 . 
as K varies. 
A0 

7. Derivation of formula: Flow in smooth pipes. Liquid flows, at a 
constant rate, through a smooth, straight pipe. Let it be required to 
find the pressure gradient G as a function of the diameter D of the 
pipe, the speed 5, density p, and viscosity ^ of the liquid. The result 
turns out to be 



8. Derivation of formula: Velocity of sound in a gas. It is easily 
shown by dimensional analysis that the velocity V of sound in a gas is 



rr 

V = constant \/- 
p 

where p and p are respectively the pressure and density of the gas. 

9. Derivation of formula: Air resistance on airplane wing. It is 
known from observations that the resistance R, which the air offers to 
the wing of an airplane, depends primarily on the shape, size L, and 
speed V of the wing and on the density p and viscosity M of air. The 
formula for the resistance R is, by dimensional methods, 



10. Derivation of formula: Period of oscillatory circuit. An oscilla- 
tory discharge is excited in a simple series circuit possessing capaci- 
tance C and inductance L. By the methods of this section the period 
t of oscillation is found to be 



= constant 



VLC. 



11. Derivation of formula: Energy density of electromagnetic field. 
Suppose that the energy density u is completely determined by the 
field strengths // and , and by the permeability /x and dielectric 
constant 8 of the isotropic medium. If E, , /* are taken as primary 
quantities, it is found that 



SOME REPRESENTATIVE RESULTS 137 

But if H, , M are taken as primary quantities, the equivalent result 



is obtained. 

12. Use of models: Water resistance to moving ship. Results 3-11 
inclusive are examples of formulas derivable by dimensional analysis. 
As a final result, we indicate how the behavior of a physical system 
may be predicted by a study of a model. For example, in naval 
architecture, the water resistance to the motion of full-sized ships is 
predicted by measuring the resistance on models. By the methods 
here considered it can be shown that the force of skin resistance P, 
which a fluid (water) offers to a ship in motion, is 

p = psv*f(, 



where the symbols have the significance: 

p = density of the fluid, 
5 = immersed surface, 

/ = a linear dimension (say the width), 
V = velocity of the ship, 

v = kinematic viscosity of water, 
P = force on the ship. 

If the same symbols with primes denote similar quantities for a 
model which has the same shape as the ship, then 



where the unknown functional relationship / is the same for both ship 
and model. If the model is run in water so that p' = p and / = v 
then the ratio P/P' is 



Now the function / is unknown but is the same in the numerator and 
denominator. We can eliminate it from the ratio if V is taken such 
that 

V V I 



138 DIMENSIONAL ANALYSIS 

IV 

Thus, if V = , the resistance on the full-sized ship is 
/ 

p/o 

P = V 2 

S'V' 2 ' 

where the numerical values of the primed quantities are obtained 
from experiments on the model. 

Inspection of the results of this section leads naturally to the 
question, What is the general theory by which they are obtained? 
Let us consider the theory in the order of its applications listed in 
48. 

50. Checking Equations. Fourier stated the principle that all 
terms of a physical equation must have the same dimensions. Let 
(?i (?2, Qn be n physical quantities (say length, viscosity, etc.), 
which are involved in some physical phenomenon. An equation, in 
(?i (?2, . . . Qn, describing a relation or motion of a physical system is 
called a physical equation. 

That the principle stated by Fourier is not necessarily true may 
be seen as follows. Consider the relations for a body falling from rest 
under the influence of gravity which involve distance fallen s, velocity 
v, and time /. We have s = \gP and v = gt, and by adding these two 
equations, we obtain 

s + v = gt + 



not all the terms of which have the same dimensions. This is a 
physical equation (according to the definition above), and obviously, 
by this definition, the principle stated by Fourier is false. 

This leads to the introduction of additional definitions and to a 
restatement of the principle. The last equation is true no matter 
how the magnitudes of the primary units involved are changed. An 
equation which has this property is called a complete physical equa- 
tion. If a physical equation is complete and if there exist no other 
relations between the quantities in the problem considered except 
those given by the equation, then all terms of the equation have the 
same dimensions, and the equation is said to be dimensionally homo- 
geneous. 

The equation of the above example is complete, but since there are 
certain other relations between the quantities involved, it is not 
dimensionally homogeneous. There exist true equations which are 
not even complete. Such equations are called adequate equations. 
Examples are given later. (See 52.) In engineering, nearly all 
equations are dimensionally homogeneous, and we thus expect all 



CHANGE OF UNITS 139 

terms to have the same dimensions or we expect to determine thfe 
reason why they do not. 

We thus make use of Fourier's statement (with the proper reser- 
vations) in eliminating errors and in remembering equations and 
formulas. Eq. (140) can be written 

IS" + T d O' + T,0 = r d o> + T.ut + /(/), 

where: 6 = angular displacement, 
/ = moment of inertia, 

T d = torque per difference of angular velocities, 
T 9 = torque per difference of angular positions, 
/(/) = applied torque. 

In view of the dimensional formulas of 48, the terms of the equation 
have the following dimensions: 



[70"] = ML 2 X T" 2 



X T-i 



[T,e] = MLT-* XLX~ = ML 2 T~ 2 , 

LJ 

[T<] = MLT ~ 2 _ XL x r-i = MI?T-\ 



[T^t] = MLT~ 2 X L X r- 1 X T 1 = ML 2 T~ 2 , 
[/(Ol = MLT~ 2 XL = ML 2 T~ 2 . 

Thus the equation is dimensionally correct. 

51. Change of Units. Change of units in all cases is reducible to a 
routine process, that is, to either the application of a simple rule or 
formula. There are two types of change of unit. In the first, merely 
the magnitudes of the measuring units are changed, for example, in 
the reduction of feet per second to miles per hour. Changes of unit 
of the first type are always possible. In the second type, not only are 
the magnitudes of the measuring units changed, but also the character 
or kinds of units are changed. For example, the primary quantities 
may be changed from length, time, and mass to length, time, and force. 
Change of units of the second type are possible only under certain 
conditions. Obviously a linear velocity, which is expressible in miles 
per hour, is not expressible also in revolutions per second. Thus a 
necessary and sufficient condition for a change of units of the second 
type is desired. 



140 



DIMENSIONAL ANALYSIS 



(a) Change of units of the first type. The following rule for change 
of units of the first type is obvious. 

RULE: Write the dimensional formula for the physical quantity 
considered, treating the dimensional symbols (or groups of them) 
as the name of concrete things. Replace each symbol (or group) by 
its equivalent in terms of the new unit of measure which is to replace it. 

EXAMPLE. Reduce Q foot-pounds per hour to ergs per second. 
Q foot-pounds (feet) (pounds) 



hour 



(hours) * 

(30.48 cm.) (444,820 dynes) 
(3600 sec.) 

centimeter-dynes 



= 3766.1 Q 



seconds 



3766.1 Qergs 
seconds 



(6) Change of units of the second type. Let it be required to change 
from a system of units in which the primary quantities are Xi, X%, 
... X n to a system whose primary quantities are FI, F 2 , . . . F n . In 
the proof for simplicity of notation, n is taken to be 3. Denote the 
physical quantity considered by Q. Suppose the dimensional 
formula 13 for Q is 

[Q] = X l ai X 2 a *X 3 a *. 
Then 

Q = h X 1 ai X 2 a2 X 3 a3 , (206) 

where &o, 01, 02, and 03 are numerical constants. Let the equations 
which relate the new primary quantities to the old be 



(207) 



where 
written, 



7/ 2 F 2 = J 
77 3 F 3 = J 
and aij(i,j = 1, 2, 3) are constants. Eqs. (207) may be 



+ 012*2 + 013*3 = 
021*1 + 022*2 + 023*3 = 
031*1 + 032*2 + 033*3 = 



(208) 



18 In 48 all dimensional formulas were seen to be products. This is true for all 
quantities of physics. 



CHANGE OF UNITS 

where # t = log X+, 

h%yi = log-ffiF,- (i = 1, 2, 3). 
The solution for x\ of (208) by Cramer's rule (26) is 



141 



#22 #23 
#32 #33 



#12 #13 
#32 #33 



#12 #13 
#22 #23 



A ' 



where A is the determinant of the system. The value of X \ then is 



Xi = (JftFi) 

Similarly, 



#22 #23 
#32 #33 



#21 #23 
#31 #33 



#21 #22 
#31 #32 



-(H 2 F 2 ) 



-(Ha F 2 ) 



#12 #13 
#32 #33 



#11 #13 
#31 #33 

#11 #12 
#31 #32 



+ 



#12 #13 
#22 #23 



#11 #13 
#21 #23 



#11 #12 
#21 #22 



(209) 

When these values of Xi, X<z, and X$ are substituted in (206) the 
physical quantity Q is expressed in terms of the new units FI, F2, 
and F 3 . 

Consider a change from a set of fundamental units consisting of 
mass in pounds (mass), length in miles, and time in hours to another 
set consisting of power in kilowatts, velocity in feet per second, and 
energy in foot-pounds. The first three units correspond to X\, X^ 
and Xz of Eq. (206), and the last three to FI, F 2 , and F 3 . We desire 
to find the magnitude of any physical quantity such as momentum 
in terms of the second set of units if it has been given in terms of the 
first set. 

Eqs. (207), which relate the new units to the old, become 



= (1 kw.) = 



33,000 ft-lb. 
0.746 min. 



33,000 


l"rM7ih ( ml Yl 


32 ' 17 lb " V5280/ 


0.746 


hr. / hr. V 


60 V3600/ 



142 



DIMENSIONAL ANALYSIS 



3.97 X 10 7 Ib. 1 mi. 2 hr.- 3 
3.97 X 10 7 XiX^X*-*, 
3600 



2 = 1 ft. per sec. = 



5280 



lb. mi. 1 hr.- 1 , 



F.-u.. l b.-f..P 2 ' 17IM "'r )Xft '1 

I sec. 2 J 



32.17 



Ib. 



- 

V5280 



\36007 
14.96 Ib. 1 mi. 2 hr.- 2 , 



or 



Ib. 1 mi. 2 hr.- 3 = 2.52 X 1Q- 8 F 1( 
lb. mi. 1 hr.- 1 = 1.47 Y 2 , 
Ib. 1 mi. 2 hr.~ 2 = 6.68 X 10~ 2 Y 3 . 

Solving for pounds, miles, and hours by Eqs. (209), we have 

ft-Ib. 



Ib. = 3.10 X lO- 2 Y 2 - 2 Y 3 l = 3.10 X 10- 2 
mi. = 3.90 X 10 Yr l Y 2 Y 3 = 3.90 X 10 6 



(ft./sec.) 2 ' 
(ft-lb.)(ft./sec.) 



kw. 



hr. = 2.66 X 10 6 Yr l Y 3 l = 2.66 X 10 6 



(ft-lb.) 
kw. 



Ni units of momentum in the first set of units are equal in the 
second set of units to 



Ib. (mass) mi. _ 



= ^ ^_ 2 



t-lb. 



hr. "* " ~~ *" ~ u ~ ~~ *' ft/sec.' 

Likewise the conversion for N2 units of kinetic energy is 
Ib. (mass) mi. 2 



(hr.) 2 



= #2 6.68 X 10- 2 Y 3 = 6.68 X 1Q- 2 N 2 ft-lb. 



And the conversion for N$ units of power is 
Ib. (mass) mi. 2 



.= ^3 2.52 X lO- 8 Yi = 2.52 X 10~ 8 N 3 kw. 



DIMENSIONAL CONSTANTS 143 

(c) Criterion for change of units of the second type. A necessary and 
sufficient condition for a change of units of the second type to be 
possible is that Eqs. (208) have a unique solution. By reference to 
26, we see that (208) has a unique solution if and only if A ?& 0. In 
the last example A = 1, and the transformation proposed was valid. 

Thus we also have a criterion as to whether certain quantities may 
be taken as primary quantities. 

EXAMPLE. Let the transformation required be the reduction of 
linear miles per hour to radians per second. Eqs. (207) become 

1 radian == 

1 second = 
Evidently 

A = o i - 

and the transformation is, of course, impossible. 

52, Dimensional Constants. A number of dimensional formulas 
have been listed in 48, but there are, of course, many more. In 
addition to dimensional formulas for physical quantities, there are 
dimensional formulas for certain constants. One of these is the 
gravitational constant. In the universal law of gravitation, 



G is a dimensional constant. (F is the gravitational force of attraction 
between two bodies of masses m\ and m% a distance d apart.) Since 



and 

M 2 



it follows that 

[G] = M 

A constant which has dimensions and which changes in numerical value 
when a change of units is made in the equation in which it is found is 
called a dimensional constant. 



144 DIMENSIONAL ANALYSIS 

In the fundamental theorem (the Buckingham TT theorem) of dimen- 
sional analysis, dimensional constants play the same role as secondary 
quantities. 

It is provable that all dimensional formulas and dimensional con- 
stants are products of powers of the primary units employed. 

It is now possible to give examples of " adequate," " complete," 
and " dimensionally homogeneous " equations. If in the two-body 
problem the unit of mass is mi + m<z and if the unit of length is the 
distance between m\ and m<i, and further if the unit of time is properly 
chosen, then the universal law of gravitation is 

(Canonical units of force) F = 

u>~ 

This equation is not true for arbitrary changes of units in L, M, and T. 
It is an adequate equation. In 50, an example of a complete equa- 
tion has been given. Also the equation 



where [G] = M 

is a complete equation. It is true for all changes of M, L t and T. It 
happens to be also dimensionally homogeneous. 

53. Introductory Problem Leading to the TT Theorem. The Buck- 
ingham TT theorem occupies the same central position in dimensional 
analysis that Newton's laws of motion occupy in mechanics. We now 
solve a simple problem which leads up to this theorem. 

Let it be required to obtain result 3 of 49. In this problem it is 
known from experience that the period of swing t depends primarily 
upon the moment of inertia /, the synchronizing torque per unit of 
angular displacement T a , and the amplitude of swing <o. Suppose, 
first, that the period of swing can be expressed as the product of powers 
of these variables, that is 

t = /*7VW, 

where x, y t and z are unknown. This is a physical equation, and we 
suppose, secondly, that it is dimensionally homogeneous. In other 
words we assume that both members have the same dimension, 
MLT or T. From 48-50 

[/] = ML*, 

[T 8 ] = MLT' 2 XL = ML 2 T~ 2 , 
<o is dimensionless. 



THE IT THEOREM 145 

By the condition of dimensional homogeneity 

[t] = [JT.W], 
or 

MLT = (ML 2 ) x (ML 2 T- 2 ^(M G LT) g . 

Thus equating the exponents of M, L, T on either side 

= x + y t 

= 2* + 2y, 

1 = - 2y. 

Thus x = ^, y = ^, and 2 is indeterminate. Hence 



where z can have any value. Moreover, if z is assigned a number of 
values, a linear combination of the corresponding terms is also dimen- 
sionally homogeneous. That is 



where the A n are constants, is also a solution. Since the series can 
represent any function we may write 



where F(0o) is any function of 0o- This is the result desired. Since 
F(0) is a dimensionless function, dimensional methods can yield no 
further information. If the last equation is divided by \/I/T, the 
left side of the resulting equation is a dimensionless product. 



In the solution of this problem, two assumptions were made. 
These assumptions were made because they are justified by the IT 
theorem. F((t> ) is determined experimentally. 

54. The IT Theorem. Let us denote by (a, 0, 7 . . . to n quanti- 
ties) a set of measurable quantities and dimensional constants. The 
total number of primary quantities in terms of which the n quantities 
(a, 0, 7 . . .) can be expressed is ra. The TT theorem is: " If the equa- 



146 DIMENSIONAL ANALYSIS 

tion/ (a, j3, 7 . . . ) = is to be a complete equation, the solution has 
the form 

F(*l, 7T2, . . .) = 0, 

where the TT'S are the n m independent products of the arguments 
a, /3, 7 . . . which are dimensionless in the fundamental units." The 
equation F(iri t 7T2, . . . ) = can be solved for a (say) obtaining 

a = /3 l y ... <t>\TT2 fl"3 . . . ). 

The use of this theorem is to predict a form of the result of a problem. 
Let us interpret the ?r theorem in terms of the introductory example. 
The complete equation desired is 

The n quantities (a, /3, y . . .) are /, /, T 8 , and fo. Two dimensionless 
products can be formed 



7T1 = t\j 
7T2 = <() 

The equation 
is in this case 



=0. 

Finally, the equation 



s 



A proof of the ?r theorem contributes very little, if any, to its suc- 
cessful application in engineering work. Consequently no proof is 
given here, but reference to Buckingham's original proof is found at 
the end of the text, Ref. 28. A number of applications of the ?r 
theorem follow, in which certain fine points in the application of the 
theorem are pointed out. The fine points are thus, perhaps, more 
easily understood than if couched in a general discussion. 

EXAMPLE 1. Let us find, by dimensional analysis, an expression 
for the resisting force R which the air offers to the wing of an airplane. 
It is known from observations that the resistance depends only on (1) 



THE T THEOREM 147 

shape, (2) size /, (3) speed v, (4) density of air p, and (5) viscosity of 
air fi. The dimensions of the last four quantities are respectively L, 

L M M 

, , and ;. Thus there are three primary quantities M, L, and T, 

J. JLj J^ JL 

and five secondary quantities R, v, /, p, and /u. By the IT theorem, 
the solution is 

i t 7T 2 ) = 0, 



where TTI and ^2 are dimensionless products. The general expression 
for TTi and ?T2 is 

7T = p T V v l*IJ, w R u , 

or 

/rx ( M\ (ML\ 

' 



This expression must be dimensionless. This demands that the equa- 
tions 

x + w + w = 0, 



3^ w 2u = 0, 



(210) 



be satisfied. We have three equations in five unknowns. Two 
unknowns may be chosen arbitrarily provided the determinant of the 
remaining three is not zero. (See last paragraph of this example.) 
For TTi let w = 0, u = 1. (These particular values are chosen to 
make the final solution simpler.) Then the equations become 

x 1, 

- 3* + y + *= 1, 



or 

x = 1, y = 2, z = 2, 
and 



For ?T2 choose = 0, w = 1. The equations become 

*= 1, 



148 DIMENSIONAL ANALYSIS 

from which x = 1, y = 1, z = 1, and 

pvl 

7T2 = 

Thus 

F (TTI, 7T 2 ) = 0, 
or 

i 2 pvi 



F\ 
If we solve for TTI in terms of ^2 and write the relation 

then 



or 

1 



which we may write in terms of a different function $(^2) as 

R = 



The function ^(^2) is, of course, dimensionless since ?T2 is. Informa- 
tion regarding <t>(^2) can be obtained by experiments on a model or 
the full-sized machine. One of the valuable uses of dimensional 
analysis, as previously mentioned, is the prediction of the action of a 
machine from the behavior of its model. This use is explained in 55 
on the principle of similitude. 

Certain restrictions on the choice of values for two of the unknowns 
in Eqs. (210) are evident. If in the determination of both TTI and TT^ 
one of the unknowns (say w) were taken to be zero, it would mean 
that M would be eliminated from both TTI and ?T2 and hence from the 
relation 

F(7ri,7T 2 ) = 0. 

Thus the result, by this choice of value for w, would be independent 
of p, and consequently wrong. There are evidently advantageous 
ways to assign values to two of the variables. In this problem, we 
are interested in eventually solving the equation for R. Hence in iri, 
u is chosen to be (1) in order that TTI is simply solvable for R. Obvi- 
ously, it is not desirable for R to enter into both TTI and 7T2, for then 
the equation 

F(TTI, 7T 2 ) = 



THE TT THEOREM 149 

is not solvable for R. Consequently in T2 take u = 0. Thus the 
choice of values for n m of the unknowns depends upon the nature 
of the problem. Other restrictions are pointed out in the following 
example. 

EXAMPLE 2. Let it be required to determine the form of the 
expression for the velocity of sound in a gas. Suppose this velocity 
depends upon the density p, pressure p, and viscosity ju. If M, L, and 
T are chosen as the three primary quantities, the dimensions of the 
secondary quantities v, p, p, and ju are respectively LT~ l , ML" 3 , 
ML~ 1 T~ 2 , ML~ l T~ l . Since there are four secondary quantities and 
three primary quantities there will be but one dimensionless product, 
TTI, which can be formed out of them. This product may be written 

7T1 = V v p r pn', 

or 



LT t 

In order that TT be dimensionless, the sums of the exponents of L, 
and T must be zero. Thus 

w = 0, 



2y z w = 0. 

One of the variables may be assigned a value. Since we shall solve 
for v, let w 1. Then 

x = |, y = - |, z = 0, 
or 

7T1 = VpKp~^. 

Since 

= 



for all values of TTI, iri must be equal to a constant. Thus 

TTI = vp^p~^ = constant. 
Solving for v we have 

v = constant */-. 

The velocity apparently does not depend upon ju. This is correct. 
Let us note a further restriction on the choice of one of the variables 
w, x, y, z. Suppose z had been chosen equal to 1 . If there is a solution 



ISO 



DIMENSIONAL ANALYSIS 



of the equations for x, y, and w the result will contain /x. But on assign- 
ing z the value 1 and attempting to solve for x, y, and w the determinant 
is seen to be 

-3 -1 1 
1 1 0=0. 
-2 -1 

By 26, there is no solution of the linear system. Consequently the 
error of supposing the result depends upon the viscosity is avoided. 

EXAMPLE 3. Consider the familiar mass and spring system pro- 
vided with damping vanes, as shown in Fig. 2. By dimensional 
analysis, let us obtain an expression for the steady-state amplitude of 
vibration A under the action of the vertical force Fsin co/. It will be 
seen that the result agrees, as far as it goes, with that obtained by 
solving the differential equation of motion. It is known that the fol- 
lowing parameters are involved in the differential equation of motion 
of an oscillating mass: 



PARAMETER 


SYMBOL 


DIMENSIONS 


Angular frequency of applied force 
Mass of body 


60 

M 


T -i 
M 


Damping constant 


kd 


MT~ l 


Peak value of applied force 


F 


MLT~ 2 


Amplitude of vibration 


A 


L 


Spring constant 


k 


MT~* 









There are six secondary quantities and three primary quantities. 
Consequently by the TT theorem there are three dimensionless products. 
The values of TTI, 7T2, TTS are obtained as follows. Let TT denote any one 
of the TT'S. Then 



or 



TT = A*o>vM*k d u F v k w , 
W = L*(T- l )*M'(MT- l )(MLT-*) 9 (MT- 2 )*- 



For this relation to be true, it is necessary that the sums of the expo- 
nents of L, M , and T to be zero, that is 

x + v = 0, 



y + u + 2v + 2w = 0. 



THE * THEOREM 151 

Three values of the unknowns may be assigned arbitrarily. 

If 

y = z = 0, x = 1, 
then 

v =- 1, u = 0, w = 1, 
and 

n - T- 
if 

* = v = 0, w = 1, 
then 

and 

= 

Finally if 

r ___A *y_-1 

at z u, w i, 
then 

y = 1, z, = 0, w=-l f 
and 

7r3 = 

In accordance with the TT theorem 

F(7Tl, 7T2, TTa) = 0, 

or 

7T1 = 0(?T2, TTa). 

Then 



Ah __ f k co^\ 
F " * Wco 2 ' i /' 



or 



The unknown function <, in this exceptional case, can be determined 
by the solution of the differential equation of motion. It is 

1 



152 



DIMENSIONAL ANALYSIS 



or 



I, TT2) = 



EXAMPLE 4. Unfortunate choice of primary quantities. Let it be 
required to determine by dimensional analysis the stiffness of a beam 
(the ratio of load to deflection) as a function of the geometrical dimen- 
sions and moduli of elasticity. The parameters involved are: 



PARAMETER 


SYMBOL 


DIMENSIONS 


Stiffness 


5 

b 
d 
E 


MT~ 2 
L 
L 
L 


Length 


Breadth 


Depth 


Young's modulus 


Shear modulus 





Since there are six secondary and apparently three primary quantities 
entering the problem, three dimensionless TT'S are expected. We have 

TT = s z l y b*d u E v n w , 



It is necessary that 



+ y + z + u v w = 0, ' 
x + Q + Q + Q + v + w = Q, 
- 2x + + + - 2v - 2w = 0. 



(211) 



When any three of the unknowns are assigned values arbitrarily, the 
determinant of the coefficients of the remaining three unknowns is 
zero since every third-order determinant formed from the array of 
numbers 



1000 1 1 
^-2000 -2 -21 

(called a matrix) is zero. This matrix is called the matrix of the coeffi- 
cients of Eq. (211). In 26 the rank of a determinant is defined. 
We now need the notion of the rank of a matrix. A matrix A is 



THE T THEOREM 153 

said to have rank r if all determinants which can be formed from the 
rows and columns of A , of order greater than r, are zero while at least 
one r-rowed determinant is not zero. 

Now Eqs. (211) have solutions. (Evidently # = l,y = 2,z = 0, 
u = 1, v = 1, w = 2 is a solution.) In fact, we have the theorem: 
Any system of m homogeneous linear equations in n unknowns whose 
matrix is of rank r < n has n r linearly independent solutions. 14 
The rank of the matrix of (211) is 2. Hence there are 6 2 = 4 
independent solutions, and hence four TT'S. But originally we antici- 
pated three TT'S! 

By inspection of the dimensions of the six quantities, it is seen 
that M and T enter only in the combination MT~ 2 . Moreover, the 
system is static, and one is led to expect that all the secondary quanti- 
ties are expressible in terms of force and length. The number of 
primary quantities chosen was too large. 

If the solution is carried out using as primary quantities L and F, 
there are four TT'S and the final result is 



* * OL\ 
l'l'Ef 



S = El<t> 

EXAMPLE 5. Use of a dimensional constant. As a final example, 
consider a problem involving a dimensional constant. Let it be 
required to find, by dimensional methods, an expression for the period 
of revolution of a planet revolving about the sun. It is supposed that 
the period depends upon the mass of the sun 5, the mass of the planet 
m, the distance d between their centers, and the gravitational constant 
G. Take as secondary quantities 5 + m, d, G, and period /. Take as 
primary quantities L, M , and T. One ?r is expected. Consequently 

TT = (S + m) x dG z t u , 

M = M*L*(M- 1 L*T- 2 )*T. 
Hence 

x z = 0, 

y + 3z = 0, 
- 2z + u = 0. 
Let u = 1, then z = |, x = J, y = -f , 

^Ori) = 0, 
14 See Refs. 17-18 at the end of the text. 



154 DIMENSIONAL ANALYSIS 

for all values of vi. Hence TTI = constant. We now have 
(5 + m)*d-*G*t l = constant, 



/ = constant 

The value obtained by solving the differential equations of motion is 

t = 



The question may arise, when does a dimensional constant enter a 
problem? The answer is, it enters whenever the general equations 
underlying the phenomena considered contain a dimensional constant. 
The universal law of gravitation seldom is used in engineering. Con- 
sequently, we are not interested in this particular dimensional constant. 

In electrodynamic problems involving the Maxwell field equations, 
the dimensional constant C (the velocity of light) enters. 

Summary. It was pointed out in the introductory paragraphs of 
this section that very little mathematical knowledge was required in 
dimensional analysis. But it is now evident that in order to apply 
dimensional analysis throughout the whole fields of physics and en- 
gineering it is necessary to know the underlying equations and prin- 
ciples of these subjects. However, dimensional analysis is applied 
to only one problem at a time, and in any particular problem the dimen- 
sional constants and the secondary physical quantities of the problem 
must be known. 

The arbitrary choice of unknowns in the homogeneous system of 
m equations in the application of the TT theorem is governed not only 
by mathematics but also by physics and engineering. The mathe- 
matical theory is stated concisely by Dickson in the single theorem, 
" Given m homogeneous linear equations in n unknowns whose 
coefficients belong to any field F and have a matrix of rank r, we may 
select r of the equations so that their matrix has a non-vanishing r-rowed 
determinant. These r equations determine uniquely r of the unknowns 
as homogeneous linear functions, with coefficients in F, of the remaining 
n r unknowns. For all the values of the latter, the expressions 
for the r unknowns satisfy the given m equations." * The system of 
Eqs. (210) is of rank three. System of Eqs. (211) is of rank two. 
The theorem tells us which unknowns may be assigned arbitrarily. 

1 



In Eqs. (211), the determinant 



1 



is not zero. Consequently, 



* Dickson, Modern Algebraic Theories, Benj. H. Sanborn & Co. 



PRINCIPLE OF SIMILITUDE 155 

u, v y w y and z may be assigned arbitrary values. The fact that the 
1 1 



determinant 







prevents the assigning of arbitrary values to 



x, u, v, and w. 

The choice of unknowns depends also upon physics. It is implied 
in the IT theorem that the matrix of the m equations is of rank m, not 
r < m. Hence the proper fundamental quantities must be chosen. 
In example 2, 54, a doubtful secondary quantity was forced to remain 
temporarily in the result by assigning the exponent unity to it. 

The choice of unknowns depends also upon engineering. The 
results must be in a form to avoid unnecessary expense in experiment- 
ing. The secondary quantity for which F(iri, ir 2 , . . . ) = 0, is to be 
solved must be simply involved in exactly one IT. In experiments 
using a model 55, certain controllable secondary quantities must 
appear advantageously in the arguments of the unknown function <J>. 

This separation of theory is for explanatory purposes only; of 
course, no logical division is implied. 

55. Principle of Similitude. The principle of similitude is, " The 
fundamental entities out of which the physical universe is constructed 
are of such a nature that from them a miniature universe could be 
constructed similar in every respect to the present universe." (See 
Ref. 29 at end of text.) Some physicists hold this principle to be an 
axiom of physics to be accepted without proof. Other physicists affirm 
it to be a natural consequence of the TT theorem. Its importance to 
the engineer is that it asserts that, as far as the principles of physics 
are concerned, a model can always be constructed. The Buckingham 
TT theorem furnishes the machinery by which we can obtain information 
regarding the action of the full-sized machine from the model. This 
has already been indicated for ships in result 12, 49. Further 
examples are now considered. 

EXAMPLE 1. The thrust T of a screw propeller of given shape 
depends upon the diameter D\ the rate of revolution n\ the speed of 
advance 5; the density p, the kinematic viscosity v of the water; and 
the acceleration of gravity g. 

If Lj M, and T are taken as primary quantities, the dimensions of 
thrust (force) are MLT~ 2 l and the dimensions of the other quantities 
are respectively (L), (T^ 1 ), (LT~*), (ML~*), (I, 2 !^), and (LT~*). 
Four dimensionless products may be formed. By the theory of 54 
the final result for the full -sized propeller is 



(f'f'f) (212> 



156 



DIMENSIONAL ANALYSIS 



Likewise for a model 



r - 

T - 



' n> D>s> 



The propeller is said to be physically similar to its model if 
Dn DS Dg\ (D_ V D'S' D'g' 



This equation is satisfied if 



Dn _ D'n' ] 
S ~ ~S r ' 

DS_ D'S' 

V V 

Dg D'g' 



' 2 ' 



(213) 



S' 



The first of (213) holds if the ratio of tip speed of blades to speed of 
advance is the same in both the machine and model. If both are run 
in water v = v', and of course g = g r . The last two equations then are 

DS = D'S', 
2)2 D >2 

~S 2 = ~S r2 ' 

These two simultaneous equations reduce to the conditions D = D' 
and S = S'. Thus in size the machine is identical to the model and 
must be run at the same speed! 

However, an approximation can be obtained. If the flow about 
the propeller is turbulent (and it is known to be so) then viscosity has 
little effect. Consequently we omit the second equation of (213). 
The last equation of (213) then gives 



D_ _ AS y 
D' " \S'J ' 



The speeds 5 and S' are called corresponding speeds. If the last rela- 
tion is substituted in 

Dg^ 

ni/~.*\~/T\i 

L 
r 



P D 2 S 2 <t> 



(Dn DS 
\S' v ' 



PRINCIPLE OF SIMILITUDE 



157 



<t> cancels and the important result is obtained that 

L - / 

T "" \D 

Thus at corresponding speeds, the ratio of thrusts from propeller and 
model is the cube of the ratio of their diameters. 

EXAMPLE 2. It is desired to predict the windage loss of a 600- 
r.p.m. synchronous condenser which has a rotor 96 in. in diameter and 
90 in. long and which is to operate in hydrogen at atmospheric pressure. 
A careful determination of the windage loss as a function of speed 
(see Fig. 25) has been made on another synchronous condenser (here- 
after called the model) of very similar design and construction but 
different size and rating. The model has a closed cooling system, but 
uses air instead of hydrogen as the cooling medium. It has a rotor 
64 in. in diameter and 60 in. long. Under normal operating conditions, 
the average temperature of the cooling air is 35 C., and it is expected 
that the hydrogen in the other machine will run at about the same 
temperature. The coefficients of viscosity at 35 C. of air and hydro- 
gen are respectively 4.05 X 10~ 7 f SUg and 2.1 X 10~ 7 -A^~. The 

ft. sec. ft. sec. 

densities at 32 F. and atmospheric pressure of air and hydrogen are 
respectively 8.09 X 10~ 2 Ib. per cu. ft. and 5.61 X 10~ 3 Ib. per cu. ft. 

Find, as accurately as possible, the windage loss at rated speed in 
the hydrogen-cooled machine from the test on the air-cooled " model. " 

It is assumed that the windage loss W depends on the following 
parameters : 



PARAMETER 


SYMBOL 


DIMENSIONS 


Diameter of rotor 


D 


L 


Physical dimensions (given by ratio) 
Speed 


LD-i 

0} 


1 
T-I 


Density of cooling medium 


p 


ML~* 


Viscosity of cooling medium 




ML~ l T~ l 









The dimensions of windage loss are ML 2 T~ 3 . By the TT theorem, there 
are three dimensionless products. Carrying out the solution according 
to the theory of the last section we have : 



158 



DIMENSIONAL ANALYSIS 



60 



or 



so 



40 



I 

J, 



7 



t 



100 MO 300 400 500 600 700 800 

R.P.M. 

FIG. 25. Windage Loss Synchronous Condenser. 

[TT] = L*(l)(T- i y(ML- 3 )(ML- l T- l ) v (ML 2 T- 3 ). 
The linear system of equations is 

x 3u v + 2w = 0, 

u + v + w = 0, 

z v 3w = 0. 



If 



u = v = w 



0, 



1, 



PRINCIPLE OF SIMILITUDE 159 

then 



If 

w = 1, u = 0, 
then 

u=-l, z = 2, # = 3, 
and 



Now W must not appear in TTS since TV is already in ?T2 and since the 
function F(ir\, T^ TTS) = is to be solved for W. Since co is the con- 
trollable factor entering into the functional relation shown in Fig. 25, 
co must appear in TTS. Accordingly let 

w = o, z = 1. 
Then 

z; = 1, w = 1, # = 2 
and 

7T3 = 



By the TT theorem, for the machine and model, we have respectively 

W = DW^TT!, 7T 3 ) 

W - 



The function will be eliminated from the ratio if - = and 
fj rs |. equation is evidently satisfied. Solving 



M 
the last equation for co' 



/> \2/ p \ /,A 
co' (" corresponding " speed of model) = 1 j I -: j I Jco 

\L> / \pv \/i / 



160 DIMENSIONAL ANALYSIS 

Since 

W = YD'> 

(1.5) 3 (2.1) (6QQ) 2 
(4.05) (180.5) 2 

But from Fig. 25, W = 1 kw. at the " corresponding f> speed, 180.5 
r.p.m. Hence, the expected windage loss in the hydrogen machine at 
rated speed is 19.3 kw. 

56. Systematic Experimentation. The value of dimensional analy- 
sis in experimental work is evident even in the simplest physical sys- 
tems. For example, suppose the time / of swing of a rotor is investi- 
gated (result 3, 49) by unsystematic experimentation. It is known 
that / depends upon /, T 8 , and <fa. By varying all parameters, one 
and then two at a time, curves are obtainable which give t as a func- 
tion of these arguments. But if, by dimensional analysis, the relation 



~\r. 



/ = + 1 F(<t>o) is obtained, it is necessary to determine by experiment 

* 1 8 

only F(<f>o), which involves varying only <o. If the desired quantity 
is a function of many arguments, it is at once evident that the saving 
made by the use of dimensional analysis is enormous. 

57. An Additional Method. In the prediction of results by use of a 
model, the method indicated in the following example is sometimes of 
use either independently or in conjunction with the ir theorem. It is 
especially applicable if the general partial differential equations denn- 
ing the phenomenon are known but cannot be integrated, as in the 
study of heat convection. Obviously, it is not feasible to discuss the 
problems involving partial differential equations at this point, but an 
indication of the method can be given by considering a trivial example 
with an ordinary differential equation. 

Suppose that we have a physical system for which the differential 



equation m ^ = / is valid. In this equation, we have mass, accel- 
dt* 

eration, and force. Suppose that we are ignorant of the dimensions of 
force. The problem is to find on what fundamental or primary quan- 
tities force depends. The differential equation for the model is 



mi - = /i. The model is similar to the machine and hence m 
* 



x = Lxi, t = Th, and / = Ffi, where If, Z,, T, and F are numbers. 
M, L, and T are known, and we wish to find F. Substituting in 



SUMMARY 161 

d 2 x 
m f the values of x, m, /, and / in terms of #1, mi, /i, and /i, we 

have 

d 2 xi _ FT 2 
dti 2 LM 

Comparing the last equation with 



we have 
FT 2 



= 1 (a dimensionless product of the quantities and hence a TT), 



LM 
or 



Now suppose mi, #1, ^ represent units of mass, length, and time. The 
Mmij Lxi, and Tt\ represent the number of units of each in the 
machine. The final result states that 

(A number representing units of mass) (A number representing length) 
(A number representing time) 2 



, , j. r r (Mass X length) _ 

and consequently the dimension of force are . By a 

(Time) 2 

very similar method, Nusselt (Ref. 30 at end of text), by a change of 
variables in the partial differential equations of machine and model, 
has determined relations which must hold between units in the heat- 
transfer equations. These relations give, in effect, the TT'S of the TT 
theorem. The method has the advantage that it sometimes gives a 
smaller number of TT'S than the ?r theorem and consequently a more 
useful result. 

58. Summary. The processes of this section are now summarized. 

(a) If an engineering equation is known to be dimensionally 
homogenous it may be checked for dimensions as indicated in 50. 

(b) A change of units of the first kind is accomplished by the rule 
of 51, part (a). A change of units of the second kind is carried out 
by mere routine substitution in Eqs. (207) and (209). 

(c) The steps in carrying out a solution by the TT theorem are as 
follows: 



162 DIMENSIONAL ANALYSIS 

1. Decide on what physical quantities and dimensional con- 
stants the unknown quantity depends. This decision rests upon 
general knowledge of the physical field in which the problem lies. 

2. Select the proper primary quantities and form the table of 
parameters as illustrated in example 3, 54. 

3. Write an expression for a general IT. Write the dimen- 
sional formula for this ir in terms of the primary quantities. Form 
the homogeneous linear system of algebraic equations. 

4. Assign values to n - m of the unknowns in accordance 
with the theory in the last paragraphs of 54. 

5. Solve for the unknown physical quantity as indicated by 
the equation 



(d) To investigate the behavior of a machine from a model obtain, 
by the IT theorem the two equations: 



a = p*i <y* 2 . . . <t>(TT 2 , TT 3 . . . ) (for the machine), 
a' = P'* 1 y* 2 - - +Cir'2 f ir'a, ) (for the model). 

If the conditions of physical similarity are satisfied, i.e., if 
ir< = IT', (i = 2, 3, . . . n m), the ratio a/a/' does not contain the 

/B\*i /v\*' 
unknown function, and a = a' ( ^ 1 ( 1 . . . . 

PROBLEMS 

1. Obtain, by dimensional analysis, all the results of 49, which have not been 
solved as illustrative examples in this section. 

2. It is desired to design a 1750-r.p.m. centrifugal pump which will deliver 
2,600,000 Ib. per hr. of mercury against a head of 85 ft. of mercury. As a first step 
in the design, a standard water pump is to be chosen which will give the desired 
performance when used with mercury. The catalogue data for water pumps include 
the following items for each pump: 

(a) operating speed Si, 

(b) head of water pumped against hi, 

(c) volume of water delivered in unit time Qi. 

In order to choose the correct pump to be used to pump the liquid mercury, 
relations between Si, hi, and Q\ may be obtained which must be fulfilled by the 
water pump which will give the desired performance with mercury. 

Considering geometrically similar pumps, the important factors are: 

(a) a characteristic dimension, such as the impeller diameter, 

(b) quantity of fluid to be pumped per unit time, 

(c) the pressure head to be pumped against, 



NATURE OF NUMERICAL INTEGRATION 163 

(d) the speed at which the pump is run, 

(e) the density of the fluid, 

(/) possibly, the viscosity of the fluid. 

Assuming that viscosity does not play an important part in the pumping process, 
obtain expressions for or relations between Q\ t hi, Si for a water pump that will give 
the desired performance when used with mercury. 

Repeat, taking viscosity of the fluid pumped into account. 

For water at 20 C. For mercury at 20 C. 

Density = 1 gram per cc. Density =13.6 grams per cc. 

Viscosity = 0.0101 -S22L Viscosity - 0.0159 



cm. sec. cm. sec. 

VI 

GRAPHICAL AND NUMERICAL METHODS OF SOLVING DIFFERENTIAL 

EQUATIONS 

Thus far, the elementary principles of the first part of the present 
chapter have led, in general, to a linear differential equation, or to a 
system of such equations, with constant coefficients. The integration 
of such equations has meant expressing the solution in the form of a 
sum of a finite number of elementary functions. The same principles 
and others also lead to equations whose solutions cannot be so ex- 
pressed. Such equations need not be linear, and the coefficients 
instead of being constants may involve both the dependent and inde- 
pendent variables. The integration of such equations, more often than 
not is very difficult, and their analytic solution is reserved for Vol. II, 
Chap. II. At present, numerical solutions may be obtained without 
additional mathematical knowledge. 

59. Nature of Numerical Integration. The solution obtained by 
graphical or numerical integration of a differential equation is a graph 
of the function which satisfies the differential equation and the initial 
conditions. For example, the numerical solution of 

L j* + Ri - E, (214) 

at 

where E, R, and L have the numerical values 100, 20, and 10, and i is 
zero at t = 0, is shown in Fig. 26. The curve shown is the solution of 
(214) only for the above values of E, jR, and L. If different values 
are assigned, all work must be repeated and this is one of the great 
disadvantages of numerical integration. We thus say that in numeri- 
cal integration the parameters are lost from the solution. 



164 METHODS OF SOLVING DIFFERENTIAL EQUATIONS 



There exists a general process by which it is always possible to 
obtain the solution (if it exists) of a single differential equation or of 
a system of such equations. This general process is usually tedious 
to apply and is employed only as a last resort. For particular problems 
of frequent occurrence, special methods are available which yield the 
desired result more easily. Accordingly, these methods are given first. 




.5 1.0 1.5 2.0 

t 



.5 1.0 1.5 2.0 
t 



Isoclines 


__ 


C 


- 


' 


c 


- .5 


/ 


c 


=1.0 


/ 


c 


=1.5 


. 


c 


"2.0 





c 


=2.5 


/ C =J.0 


1 


c 


=3.5 


. 


c 


*4.0 


, 


c 


=4.5 


/ 


c 


=5.0 


i 


c 


5.5 


1 


c 


=6.0 


/ C =6.5. 


/ 


c 


-7.0 




c 


=7.5 


1 


c 
c 


'8.0 
=3.5 


li 


c 


=9.0 


\ 


c 
c 


=9.5 
10.0 



i -5.0 
i=4.75 
i =4.50 
i =4.25 
i =4.00 
i=3.75 
i =3.50 
i=S.25 
t s&00 

i=2.50 
i'2.25 
i =2.00 
i=1.75 
i1.50 
i 1.*5 
t=1.00- 
t .75 
i .50 
i- .25 
i= 



FIG. 26. Numerical and Graphical Integration of Differential Equations. 

di 

60. The Differential Equation =/(i; t). The graphical and 

at 

numerical methods in this case are very simple. They are based on 

di 

the fact that is the slope, at the point (t, i), of the curve which is the 
at 

solution of the differential equation. Consequently, if at a sufficient 
number of points (/, i) the values of /(i; /) be computed the direction 
elements may be drawn at each of these points. If enough direction 
elements be put in, the directions of the solutions at any point are 
given and the curves may be sketched which are the infinitude of solu- 
tions of the differential equation. Among these curves there will be 
one which satisfies the initial condition i = io(/o), and is the solution 
of the physical problem considered. 

Graphical solution. A convenient way of carrying out a graphical 
solution is known as the method of isoclines. A curve obtained by 
setting /(i; /) = C (a constant) is called an isocline because at every 

di 
point of such a curve the slope = /(i; /) = C is a constant. Let the 



FIRST ORDER DIFFERENTIAL EQUATIONS 



165 



= 10 2i, for the initial condition i = for 



family of curves be drawn for various assigned values of C. For 
instance, at every point on the isocline /(i; t) = 1 (or c\) the tangent 
makes an angle of 135 (or arc tan" 1 ^) with the positive #-axis. By 
inserting a sufficiently large number of isoclines the tangents to the 
solution and hence the solutions themselves may be drawn. 

EXAMPLE 1. Let it be required to obtain the graphical solution of 

di _ E - Ri 

Le ' f ~dt = 1 
t = 0. 

The isoclines 10 - 2i = C(C = 0, .5, 1, 1.5, 2, . . ., 9.5, 10) are lines 
parallel to the /-axis (Fig. 26). The third column from the right 
indicates the slopes at which the directional elements cut the isoclines. 
Beginning at the origin and inserting successively the directional ele- 
ments whose slopes are 10, 9.5, 9, 8.5, 8, . . ., 0, we have the curve com- 
posed of arrows which are tangents to the solution of the differential 
equation. The solution itself is now easily drawn. The correct solu- 
tion, found by analytic integration, passes through the centers of the 
small circles shown on the broken line curve. 

Numerical solution. A convenient way of carrying out a numerical 

di 
solution of = f(i\ /) is as follows. Suppose i = <t>(f) is the solution 

di dt 

of -j = f(i] t) satisfying the initial condition i = io(/o). Let /o, /it fe, - . . 



dt 
and io, ii, i'2, 



denote the quantities shown in Fig. 27. Let 




FIG. 27. 

tk = h (k = 0, 1, 2, . . .). If h and the curvature of i = <t>(f) are 

d<t> 

small, then the value of within the interval // is given approximately 

at 



d 



166 METHODS OF SOLVING DIFFERENTIAL EQUATIONS 

From this equation 

<t>k + h 
at 



Let the average value 



d<t>\ 

be 

dt lave. 



dt 



approximated by 

+ -77 



d4 

dt 



dt 



where 
at 



and 



d*\ 



fc+i 



are respectively the slopes of * = <t>(t) at the 



points (tk, 4) and (/ t+ i, i k +i). Thus 



, dt 



(215) 



In constructing the solution, the value io = </>o is known from the 



.... ,. . , di 

initial condition and - 

dt 



d<> 



is easily computed from the differ- 

o 

d<t> 

is estimated from the slope of 



ential equation. The value of 

dt i 

the tangent at (/o, io). The value of <i is then computed from (215) 
by letting k = 0. If this value of <i when substituted for i in the 

di di 

right member of -7 = f(i; t) renders 



dt 



dt 



dt 



, then the value of 



is sufficiently accurate and the point (/i, i\) has been located on the 



curve. Next 
dt 



is estimated and fa calculated by (215). fa is 



di 
checked in the differential equation if 



dt 



d<t> 
~dt 



continued for is, 1*4, 



The process is 



di 



EXAMPLE 2. Let us solve, by the method just given, - = 10 2i 

dt 

subject to the initial condition i = at / = 0. Then, 



~ di 

- dt 



If h - 0.1 by (215), 



d<t> 
dt 



10. 



FIRST ORDER DIFFERENTIAL EQUATIONS 

d<t> 
From the slope of the tangent at t 0, estimate = 9. 



dt 



<tn = 0.95. Substituting fa = 0.95 in 

di , di 

- = 10 - 2*. - 



167 
Then 



dt 



dt 



8.1. 



The estimated slope is too large. Try 



dt 



= 8.2. Then 



di 



Next 



8.2. Thus 



di 
dt 



djt> 
Tt 



8.18. 



Continuing the process, we have the table: 



/ 


d<$>\ 
Estimated 
dt \k+\ 


<t>k+l 


-1 

dt\k+i 





10.0 




10.00 


0.1 


8.2 


0.91 


8.18 


0.2 


6.7 


1.65 


6.70 


0.3 


5.5 


2.26 


5.48 


0.4 


4.5 


2.76 


4.50 


0.5 


3.7 


3.16 


3.68 


0.6 


3.0 


3.49 


3.02 


0.7 


2.5 


3.77 


2.46 


0.8 


2.0 


3.99 


2.02 


0.9 


1.6 


4.17 


1.66 


1.0 


1.3 


4.32 


1.36 



The graph is the second curve in Fig. 26. 
EXAMPLE 3. Find the solution E = <(/) of 

dE _ e - ri _ e - rF(E) 
dt ~ K X 10- 8 ~~ K X 10- 8 



0.91 



(216) 



subject to the initial conditions i = for t = and where i = F(E) 
is given, not by an equation, but by the curve of Fig. 28. This 
problem arises in determining the time rate of build-up of the armature 
voltage of a separately excited direct-current generator, when a con- 
stant voltage is applied to the field. 



168 METHODS OF SOLVING DIFFERENTIAL EQUATIONS 

The symbols have the following significance: 

E = armature voltage, 
e = constant voltage applied to field, 
i = field current, 
r = resistance of the full circuit, 
K = ratio of field flux linkages to armature voltage. 

The constant K depends upon the construction of the generator, as 
well as the speed, which is assumed constant, and i = F(E) is the 
saturation curve of the generator. 




2 It 6 8 10 12 14 16 

FIG. 28. 

Assume values of 0, 1, 2, 3, ... for i, read E from the curve 

dE 
i = F(E), and compute for i = 0, 1, 2, 3, . . . . This gives values 

dE 

of E and for a number of points. Since from physical considera- 
dt 

tions, out of which the problem arose, E and i are both continuous 

dE 

functions of /, we can obtain at points whose abscissas fo, h, fe, 

at 

are as near together as we please by taking 4+i ik h sufficiently 
small where k = 0, 1, 2, .... 

Since =</>(/) passes through the origin and we know its slope at 
to, ti t t2, fe, . . we have shown how to obtain its graph (Fig. 29) and 
the problem is theoretically solved. 

Instead of plotting as a function of time, a different procedure 

dE 

is sometimes followed. From Eq. (216), is determined as a func- 

at 

dE dt 

tion of E and then the reciprocal of -, or , is plotted as the func- 

dt ah, 

tion of E. (See Fig. 30.) 



FIRST ORDER DIFFERENTIAL EQUATIONS 



169 



Thus 



dt_ _ K X 10~ 8 
dE ~ e - rF(E)' 

._ . " 10- 8 

t 



r o e - rF(E) 



dE. 



FIG. 29. 




FIG. 30. 

Evidently the time taken by the generator to attain the voltage 
E = D\ is the area EoDiD\a on the figure, and this may be found by 
numerical integration, for example by the trapezoidal formula 



f(E)dE = 



where 



E k+1 -E k = h (k = 0, 1, ... n - 1) 
and /* is the ordinate of the curve at E = *. 



170 METHODS OF SOLVING DIFFERENTIAL EQUATIONS 
61. The System of Differential Equations: 

~Tt ' 



(217) 



Eqs. (217) may be integrated numerically by the general method to 
be explained in 63-65, or they may be easily integrated graphically 
in the following manner. 

If the second of (217) is divided by the first there results 



dy 

dx 



/(*oO 



= h(x,y). 



(218) 



Eq. (218) is integrable by the method of isoclines, and the result is a 
functional relation (given by a graph) between x and y. Let this 
relation be denoted by y = fi(x) or x =/2(y). Suppose the initial 
conditions for the system (217) are XQ = x(to) and yo = y(to). From 
(217), x and y can be expressed as functions of t by the relations, 



(219) 



r' r d x r dx 

I dt = I = I , 

4 J* a f(x,y) Jf(x,Mx)}' 

jr' dt = ^_ = jT-^ 



The integrals in Eqs. (219) can be evaluated by the trapezoidal rule. 

d 2 y 

EXAMPLE. Integrate = 0.9 sin y subject to the initial con- 
at 

ditions y = 0.925 (radian) and -~ = for / = 0. The differential 
equation may be replaced by the system 



. 
--0.9-.rn* - = 



(220) 



d 2 y 



Eqs. (220) are called the normal form of jj = 0.9 sin y. The 

method of reducing any differential equation of the nth order, or any 
system of differential equations, to the normal form is explained in 
64. 



THE SYSTEM OF DIFFERENTIAL EQUATIONS 



171 



To carry out the graphical solution of (220), divide the second by 
the first and obtain 

dy __ 0.9 sin y 

doc OP 



(221) 



i.t 




' -.04 o .ot .os .it 
FIG. 31. Method of Isoclines. 



Eq. (221) corresponds to (218). The isoclines are 

0.9 - sin y 1 



C 



or 






- sin- 1 - (* - 0.9C). 



172 METHODS OF SOLVING DIFFERENTIAL EQUATIONS 

If C is assigned the values 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.5, 1.7, 2, ... 100, 
we obtain the isoclines shown in Fig. 31. The slopes on the isoclines 

x :.= 0, y = sin" 1 (x 0.18), and y = sin" 1 (x 0.36) are 

respectively 0, 0.2, and 0.4. Continuing, we draw the oval curve in 
Fig. 31. In this particular problem, we can obtain the equation of 
the oval curve by integrating (221). This is unusual, however. The 
analytic solution of (221) satisfying the initial conditions of the prob- 
lem is 

x = [l.Sy + 2 cos y - 2.86867]*. 



1.5 



0.9 



O.I 



. 0.7 



-O.t 



2 t 6 8 10 IS 

Time in Sec. 



9 4 6 8 10 1 

Time in Sea. 

FIG. 32. Results Simplified Hunting Equation. 



The curves expressing x and y as functions of t are found by carrying 
out numerically, by the trapezoidal rule, the integrations indicated in 
Eqs. (219). From the second equation in (219) 



= r ^. 

^0.925 X 



The values of x and y in this integral are the abscissa and ordinate 
of the point P(x> y) on the oval curve in Fig. 31. The numerical 
integration is given in the following table. 



THE RADIUS OF CURVATURE METHOD 



173 



y 


X 


1 

X 


av - 

X 


Ay 


A; 


t 


0.925 















0.9255 


0.0101 


99.01 


99.01 


0.0005 


0.05 


0.05 


0.927 


0.0201 


49.75 


74.38 


0.0015 


0.11 


0.16 


0.934 


0.0421 


23.75 


36.75 


0.007 


0.26 


0.42 


0.942 ' 


0.0572 


17.48 


20.62 


0.008 


0.16 


0.58 


0.960 


0.0798 


12.53 


15.00 


0.018 


0.27 


0.85 


0.995 


0.1065 


9.39 


10.96 


0.035 


0.38 


1.23 


1.030 


0.1223 


8.18 


8.78 


0.035 


0.31 


1.54 


1.065 


0.1317 


7.59 


7.88 


0.035 


0.28 


1.82 


.100 


0.1361 


7.35 


7.47 


0.035 


0.26 


2.08 


.135 


0.1364 


7.33 


7.34 


0.035 


0.26 


2.34 


.169 


0.1329 


7.52 


7.42 


0.034 


0.25 


2.59 


.204 


0.1256 


7.96 


7.74 


0.035 


0.27 


2.86 


.239 


0.1141 


8.76 


8.36 


0.035 


0.29 


3.15 


.274 


0.0972 


10.29 


9.52 


0.035 


0.33 


3.48 


1.309 


0.0718 


13.93 


12.11 


0.035 


0.42 


3.90 


.320 


0.0602 


16.61 


15.27 


0.021 


0.32 


4.22 


1.330 


0.0479 


20.88 


18.74 


0.010 


0.19 


4.41 


1.340 


0.0288 


34.72 


27.80 


0.010 


0.28 


4.69 


1.344 


0.0156 


64.10 


49.41 


0.004 


0.20 


4.89 


.345 


0.006 


166.67 


115.38 


0.001 


0.12 


5.01 


.345 


-0.006 


-166.67 


-166.67 


-0.0008 


0.13 


5.14 


.344 


-0.0156 


-64.10 


-115.38 


-0.001 


0.12 


5.26 


.340 


-0.0288 


-34.72 


-49.41 


-0.004 


0.20 


5.46 


.330 


-0.0479 


-20.88 


-27.80 


-0.010 


0.28 


5.74 


1.320 


-0.0602 


-16.61 


-18.74 


-0.010 


0.19 


5.93 


1.309 


-0.0718 


-13.93 


-15.27 


-0.021 


0.32 


6.25 



The functional relations shown in this table between y and t and 
between x and t are shown also in Fig. 32. 

62. The Radius of Curvature Method. Equations of the form, 

dy 



or 



y" = f(y,y';t) (222) 

can be integrated graphically as follows. Dividing Eq. (222) by 
(1 -j. - y '2) i we have 

y" 



The function g(y, y';f) is thus the curvature of the solution of Eq. 
(222). Consequently, the radius of curvature R is 

R = , 1 , , v (223) 

g(y,y;f) 



174 



METHODS OF SOLVING DIFFERENTIAL EQUATIONS 



From the example of 61, it is evident that through every point 
(/, y) of the /y-plane there passes an infinitude of solutions of a second- 
order differential equation. There is, however, only one solution 
having a prescribed direction through each point of the plane. 

Beginning at a prescribed point PO in a prescribed direction 

dy 

= DO draw through PO a circle Co whose radius of curvature RQ 

at o 

is computed by (223). (See Fig. 33.) At PO, y, y', and t are all 




0.9 



FIG. 33. Integration by Curvature Method. 

known from the initial conditions of the problem. At a point PI 
near PO and on the circle Co determine graphically y, y' t and /. Sub- 
stituting these values at PI in (223), compute the radius of curvature 
RI for the circle C\ through PI and tangent to Co at PI. Continuing 
this process, we obtain the points PO, PI, P2, ... on the solution of 
the differential equation. If the process is repeated by taking PoPi, 
P2P2 equal to half their first values and the curve of the solution 
is but slightly changed the solution is sufficiently accurate. 

EXAMPLE 1. In order that the different methods of this section 
can be easily compared, obtain the solution of the example of 61 by 
the method of 62. 



THE RADIUS OF CURVATURE METHOD 



175 



To obtain convenient values of R for graphical work it is advan- 
tageous to make a change of independent variable in the equation 



= 0.9 sin y. 



dt* 



Accordingly, let 5r = /. Then the differential equation to be solved 
graphically is 



-75 = 25(0.9 - siny). 

The radius of curvature at the point (r, y) of the solution of the 
differential equation is 



R = 0.04 



0.9 sin y 



Let the coordinates of P be (0, 0.925). The values of y, y', and R at 
the points PO, PI, P2 are found, by the method of this article, to 
be as follows: 



y 


r 


y' 


Rtf = 0,1, ...,22) 


0.925 








0.39 


0.927 


0.046 


0.080 


0.40 


0.930 


0.072 


0.175 


0.42 


0.940 


0.120 


0.260 


0.48 


0.950 


0.153 


0.350 


0.55 


0.960 


0.181 


0.420 


0.63 


0.980 


0.220 


0.500 


0.80 


.000 


0.255 


0.570 


1.03 


.025 


0.295 


0.690 


1.57 


.050 


0.337 


0.750 


2.39 


.075 


0.368 


0.790 


4.04 


.100 


0.400 


0.800 


9.54 


.125 


0.431 


0.800 


-35.00 


.150 


0.462 


0.800 


-6.62 


.175 


0.495 


0.790 


-3.63 


.200 


0.526 


0.760 


-2.47 


.225 


0.562 


0.700 


-1.78 


.250 


0.600 


0.640 


-1.37 


.275 


0.640 


0.550 


-1.05 


.300 


0.685 


0.500 


-0.88 


.325 


0.744 


0.410. 


-0.73 


.350 


0.820 


0.260 


-0.58 


.360 


0.880 


0.150 


-0.53 



176 METHODS OF SOLVING DIFFERENTIAL EQUATIONS 

The construction is shown in Fig. 33. If the time is to be expressed 
in seconds the numbers along the r-axis must be multiplied by 5. 

63. Preliminary Ideas for the General Method of Numerical 
Integration. If the differential equation or the system of differential 
equations is not too complicated, the methods so far given will give a 
graphical or numerical solution. But systems are occasionally ob- 
tained where recourse must be had to the general method of numerical 
integration referred to in 59. This general method is laborious and 
such that one significant error made at any step invalidates all the 
work following it. Consequently it is best to develop formulas, tables, 
and methods of procedure which reduce the labor to mere routine in 
order that full attention may be given to the actual numerical work. 
The use of these formulas and tables will become apparent in 65. 

We need first the equation of a polynomial which approximates a 
given smooth curve over a given interval. Suppose that the equation 
of the curve is y = f(t) and that the curve passes through the points 

ft*yn), (/n-ljn-l), (/n-2jn-2), (/l,yi), where 

/,+i ti = h (i = n 1, n 2, . . . , 1). 

Let the approximating polynomial of the nth degree be written in 
the form 

F(f) = ao + ai(/ / n ) ~\- a%(t -~ t n )(t t n -i) 

+ CL3(t OO ^n-l)(J ^n 2) + . . . 

+ a n (t - t n )(t - / n -i) ...(/- /]). (224) 

The coefficients ao, ai, . . . , a n are determined by the conditions 

F(tn) = y f F(/n-i) = yn-i, . . . F(ti) = yi. 
Applying these conditions to (224), we have 
= y n = a , 

n-l - /n), 
-2 - /n) 4 

etc. 

From these equations, 

ao = y n , 

y n - y n -i 
ai = ; , 

(225) 

a * = ~ yj w 

etc. 



PRELIMINARY IDEAS 177 

We need next difference tables. By means of differences, Eq. (225) 
can be more conveniently expressed. Suppose that the values of y 
(say yo, yi, y 2 , . . .) corresponding to the values / = / , /i, te, /a, ... 
are known. Further let 

/,-+! - /< = h for i = 0, 1, 2, 3, ____ 

Then the first differences of the function y = /(/) (written Aiy) are 
defined to be 



= yi - yo, 
= y2 yi, 
= ya yz, 
etc. 

The second differences are the first differences of the first differences: 

= y 2 2;yi + ^o, 



etc. 



Similarly, third and higher differences are formed. A difference table 
for the function y = /(/) is written : 



/ y Aiy A2y Asy A4y 

to yo 

h yi 

/2 y2 

/3 ya Aiy 3 A 2 ys Asys 

etc. 

To obtain from this table, say, A 2 y4, find in the same row the entry in 
the first column to the left, Aiy4. From the last-mentioned element 
subtract the element above it, thus 



178 METHODS OF SOLVING DIFFERENTIAL EQUATIONS 
In view of the difference table, Eqs. (225) become 



, 



and Eq. (224) is 

~ 



02 = 

etc., 



~ A.) 



fc-i) ..-(<- <i). (226) 
Last, approximate values of the integrals 

ri /*'n /"fiM-l 

/*, / /(0#, and / /(/) 
-^-* *^rv-l *^n 

are required. 

Formula (226) is called Newton's formula for backward interpola- 
tion. At the points (/ n , y), (^n-i, ^n-i), . . . (/i, yi), Eq. (226) gives the 
values of the function y /(/) exactly. At intermediate points of 
the intervals (t n / n -i), (^n-i - ^1-2)1 and (/ n _2 /n-a) the formula 
is used for interpolating values of /(/). Moreover, it is used for extra- 
polating values to y = /(/) in the interval t n +i t n provided this 
interval is sufficiently small. 

Consequently, to find the approximate value of the last three 
integrals, replace /(/) by F(t). The resulting integrations are easily 
carried out by the substitution / = t n hv. Formula (226) then 
becomes 

F(f) = y 
Then 




_ 2 ) + .. 

. . .) (227) 



REDUCTION OF SYSTEMS OF EQUATIONS 
Likewise, 

= h(y n - \ A,y B - ^ A ^ n - ^ A 3 ? n -...), (228) 



/n+l 
f(f)dt = h(y n + \ ^y n + A A^n + f A 3 ;yn + . . .). (229) 
n 

64. Reduction of Systems of Equations to the Normal Form. The 

general theory of numerical integration is applicable only to differential 
equations in what is known as the normal form. This form consists 
of a system of simultaneous equations, the left members containing a 
single first derivative, while the right members contain no derivative. 
The number of equations in the normal form of the system equals the 
order of the system. (See 18.) The reduction to the normal form 
is merely a routine process. One new dependent variable must be 
introduced for each differentiation of order higher than the first which 
occurs. The process is illustrated as follows: 

EXAMPLE 1. The differential equations of the motion of a pro- 
jectile, under proper assumptions, are 

d 2 x dx 

dt 2 " * dt' 



= K P. 

7fO ^ 7, &t 

at* at 

where A; and y are the coordinates of the projectile, / is the time, and 

., H(y)G(V) 

k ~ c 

The constant C is the ballistic coefficient dependent upon the shape of 
the projectile, H(y) is a function of the height of the shell above 
ground, and G(V) is a function of the velocity. 

Reduce these equations to the normal form. Let 
x = xi, y = xs, 

dx _ dy 

Then 



180 METHODS OF SOLVING DIFFERENTIAL EQUATIONS 
dx 2 



The last four equations are the normal form of the two second-order 
differential equations of motion of the projectile. 

65. General Method of Numerical Integration of Diff erential . 
Equations. The notation for the general theory for n normal equations 
is complicated. Accordingly, the exposition is given for the system 



dx 


f " *#*> 
x = XQ, y = y , / = 0. . 



(230) 



The method is easily extended to any number of equations. The 
exposition does not contain any proof that the processes employed 
actually converge to the solution required. Such a proof exists, but 
we are interested only in the steps required to obtain a solution. The 
carrying out of a numerical solution consists of two parts: (-4) starting 
the solution, (B) continuing the solution. Each of these parts con- 
sists of several steps. 

(A) Starting the solution. 

1. The first step of the solution consists in choosing an increment 
h of /, which will be the time-interval between successive desired points 
of the solution. No definite rule for determining a good size for h can 
be given, but a method for ascertaining whether any chosen value is 
too great or too small will appear later. 

dx dy 

2. In the next step, we calculate and -f 1 at / = from (230). 

at at 

. 3. We now assume that, over the small interval h, the first deriv- 
atives are approximately constant, and obtain 

*i (1) = *o + A/o, 1 

(231) 

yi (1) = yo + Ago. J 



GENERAL METHOD OF NUMERICAL INTEGRATION 181 

The superscript (1) indicates that this is the first approximation, and 
/o = /(*o, yo; 0), go = g(# , yo\ 0). 

4. From the approximate values of x\ and yi given by (231), we 
calculate /i (1) and gi (1) from (230). 

5. A better approximation to x\ and y\ may now be obtained by 
assuming that the average slope from point to point 1 is the average 
of the slopes at these points. This gives 



, 

yi > = :yo + /Ko + gi (1) ). I 

6. Using the second approximations given by (232), new approxi- 
mations to /i and g\ are calculated. 

7. From the new approximations /i (2) and gi (2) , third approxima- 
tions of x\ and yi are calculated : 

^=^o + ^(/o+/i (2 >),[ 



8. From x\ (3) and 3>i (3) the values of /i (3) and gi (3) are calculated 
from (230). 

9. Steps 3 to 8 are now repeated for the second interval, the results 
being # 2 (3) , 3'2 (3) ,/2 <3) , 2 (3) . At this stage (# , yo), (*i, yi), and (x 2 , y*) 
are all known; the first pair exactly and the others approximately. 
We now correct these approximations. 

10. We construct difference tables for/ and g: 



to /o go 

h fi AI/I gi 

fe /2 Ai/2 A2/2 2 

11. Now the exact values of x\ and y\ are 



(234) 



xi = XQ + I fdt, yi = yo + I gdt. (235) 

(This is seen to be true, since, if 

x = XQ + I fdt, and y = yo + I fdt 

*0 *0 

are substituted in (230), the differential equations are satisfied. The 
values of x and y given by the last two equations are for any time /. 



182 METHODS OF SOLVING DIFFERENTIAL EQUATIONS 

When / = /i, then x = xi and y = y\.) We apply (227) to find cor- 
rected values of x and y. In (227), let n = 2, y = /, and obtain 

}dt = A(/ 2 - | Ai/2 + A A 2 / 2 ) (236) 

and a similar result for the integral of g. 

12. By substituting (236) into (235), we obtain #i (4) and y\ ( *\ from 
which and (230) we calculate /i (4) and gi (4) . 

13. Next, repeat steps 9, 10, and 11, if necessary. 

14. Since the exact values of X2 and y^ are 

x 2 = xi +J fdt, y 2 = yi +J gdt, (237) 

we apply (228) with n = 2, obtaining 



//i 



fdt = A(/ 2 - | Ai/2 - -A- A,>/ 2 ) (238) 



and a similar result for g. 

15. By substituting (238) into (237), we obtain new approxima- 
tions to X2 and y%, from which we calculate new values of /2 and gz. 

16. The new values of /2 and g2 and their differences may be used 
to recalculate xi and y\ ; but if the value chosen for h is as small as it 
should be this will be unnecessary, and it may be assumed that the 
values at points 0,1,2 are correct and that the solution is fully started. 

(B) Continuing the solution. 

1 7. The new values of/ and g are used to construct a new difference 
table similar to (234). 

18. By an application of (229) with n = 2, values of x$ and y$ are 

extrapolated : 

h 

fdt = x 2 + h(f 2 + \ A,/ 2 + A A 2 /o) .... (239) 

19. From the values #3 (1) and ^3 (1) , calculate values of /3 (1) and 
g3 (1) from (230), and append these and their differences on to the table 
of step 17. Third differences now enter. 

20. Apply (228) with n = 3 to correct #3 and ^3, obtaining 

* 3 (2) = *2 + / /d/ = *2 + A(/3- J Ai/3 & A2/3-.& A 3 / 3 ) .... (240) 

*4 

21. Repeat steps 19 and 20 if necessary. 

22. The above steps are now repeated for the next interval, and 
the solution is well under way. 



GENERAL METHOD OF NUMERICAL INTEGRATION 183 

From the preceding steps it will be seen that Eq. (227) is used to 
check the next to the last point, (228) to check the last point, and 
(229) to extrapolate to the next point. 

Tables of differences for x and y are unnecessary, but are useful in 
checking the work. If these differences become irregular it is probable 
that an error has been made. 

If the third differences of / and g are approximately constant the 
interval h may be doubled. On the other hand, if they are quite 
irregular h should be halved. This may be done by interpolating 
points half-way between the calculated points by means of (226). 

If several trial calculations (step 21 or steps 7 and 8 above) are 
necessary before an accurate value is obtained, the value of h should 
be halved as explained above. 

Example on the general method. In order that the general method 
may be compared with the methods of 61-62, we again solve the 
example of 61. Eqs. (230) for this example are 



dx 

= 0.9 - sin y =/(*,y;0, 

at 



(241) 



= % = g(x,y\t\ 

yo = 53 = 0.925 radian, XQ = 0, / = 0. 

The numbers of the statements below refer to the corresponding 
numbers of the steps of 65. Many steps may be omitted unless exces- 
sive accuracy is desired. 

1. h = 1 second. 8. /i (3) = 0.0719 

o f mm A n & = - 0866 

2. /o = 0.1014, go = 

3. jci (1) = 0.1014 9. x<2 ( " = 0.1585 
;yi (1) = 0.925 y 2 (1) = 1.062 

4 fl ci, . oiOU /* (1> =a 267 

CD I oiou 2(1> =ai58S 

gl U - 1U14 ^(2) = 0.1359 

5. ^i (2) = 0.1014 y 2 (2) = 1.0983 

yi (2> = 0.9757 / 2 (2) = 0.0096 

6 *> -00719 ^ (2> =0 ' 1359 

(2> I oiOU " 2(3) ==al274 

ft :y 2 (3) = 1.0869 

7. JCi (3) = 0.0866 jf 2 (3) = 0.0148 

3^(3) = 0.9757 g 2 (3) = 0.1274 



184 METHODS OF SOLVING DIFFERENTIAL EQUATIONS 

10. Difference table for/ 
/ 
/o 



A 2 / 



0.1014 

0.0719 -0.0295 

0.0148 -0.0571 -0.0276 



11. Difference table for g 



t 

to 







tl 


0.0866 0.0866 




fe 


0.1274 0.0408 -0.0458 




12. *i> 


= + [0.0148 + | (0.0571) - & (0.0276)] 




= 0.0889 




yl <4) 


= 0.9250 + [0.1274 - f (0.0408) - -^ 


- (0.0458)] 




= 0.9721 




14. *2 (4 > 


= 0.0889 + [0.0148 + % (0.0571) + -^ 


r (0.0276)] 




= 0.1346 




y 2 (4> 


= 0.9721 + [0.1274 - (0.0408) + ^ 


- (0.0459)] 




= 1.0829 




It 


is unnecessary to recalculate f-z and g 2. 




18. * 3 (1) 


= 0.1346 + [0.0148 - \ (0.0571) - ^ 


- (0.0276)] 




= 0.1094 




y 8 U> 


= 1.0829 + [0.1274 + i (0.0408) - -j^ 


- (0.0458)] 




= 1.2117 




19. / 3 (1) 


= - 0.0362 




g3 U) 


= 0.1094 




/ 


/ Ai/ A 2 / 


A 3 / 


to 


0.1014 




tl 


0.0719 -0.0295 




fe 


0.0148 -0.0571 -0.0276 




fa 


-0.0362 -0.0510 0.0061 


0.0337 


; 


g Aig A 2 g 


A 3 g 


/o 







<i 


0.0866 0.0866 




fe 


0.1274 0.0408 -O.C458 




/3 


0.1094 -0.0180 -0.0588 


-0.013C 



SUMMARY 185 

20. 

* 3 (2) = 0.1346 - 0.0362 + (0.0510) - -^ (0.0061) - ^ (0.0337) 

= 0.1220 
^ 3 (2) = 1.0829 + 0.1094 + | (0.0180) + -^ (0.0588) + ^ (0.0130) 

= 1.2067 

The solution is now started, and additional points are simply found. 
If the four points obtained are plotted in the graph of Fig. 32, they 
are seen to give very approximately the results previously obtained. 

66. Summary. The methods of 60-63 should be tried first. 
If the system of equations is too complicated, the use of the general 
method of 65 may be necessary. If the problem is one based on 
electrical phenomenon and if the dependent variables are currents or 
voltages, the results obtained by numerical integration contain only 
the information available from an oscillogram. If the problem is a 
mechanical one and if the system of differential equations contains 
many parameters (say 12), this requiring several integrations, re- 
course may be had to the differential analyzer. 15 The results in this 
case are a book of curves; the parameters have been lost from the 
solution. However, the engineer is interested in the behavior of 
systems for various values of parameters. Formulas are needed. 
Consequently analytical methods are essential. The differential equa- 
tions of problems 4 and 5 below, along with many others, are solvable 
analytically. (See Ref. 6). Appreciation of analytical methods is 
enhanced by one or two attempts at numerical integration where the 
general method of 65 must be employed. 

PROBLEMS 

1. The rate of build-up of the armature voltage of a separately excited d-c. gen- 
erator with field coils in parallel is given by the relation 

dEA __ (E - ipRp) 
dt KX Np ' 

where EA - armature voltage, 

E = a constant voltage suddenly applied to the field, 
ip field current per winding, 
Rp field resistance per winding, 
Np = number of field turns per winding, 
K = a constant depending on design. 

If the saturation curve is available and the machine constants are known, the 
value of dEA/dt may be obtained for any value of EA by substituting the value of ip 
corresponding to EA (from the saturation curve). 

16 See Journal of the Franklin Institute, October, 1931. 



186 METHODS OF SOLVING DIFFERENTIAL EQUATIONS 



The reciprocal of dEA/dt, namely, dt/dEA, may then be plotted as a function 
of EA- The area under this curve is evidently t\ i.e., the time required for the 
armature voltage to build up to any value is the area under the curve up to that 
value of armature voltage. This area may be found by numerical integration. 



Given that E 125 volts (applied at 
RF = 6 ohms, 
NF = 750 turns, 
K = 14,140 X 10~ 8 , 



0), 



and that the saturation curve is determined by the following data: 



ARMATURE 
VOLTAGE 


FIELD CURRENT 
PER POLE 


ARMATURE 
VOLTAGE 


FIELD CURRENT 
PER POLE 


65 


1 


305 


7 


130 


2 


316 


8 


189 


3 


325 


9 


235 


4 


333 


10 


267 


5 


341 


11 


289 


6 







plot the curve of armature voltage as a function of time. 



2. Solve the equation 



o/ 2 



0.9 sin y by the method of 62 and without mak- 



ing any substitution for change of scale. 

3. A sphere moving with a small velocity in a fluid experiences a resisting force 
F = 6-n-prv. For any velocity the resistance is F = %irpr' 2 v 2 C w (R), where r is the 
radius of the sphere, v its velocity, ju the viscosity of the fluid, p its density, R = prv/v, 
and C W (R) is a function of R given below. The differential equation describing the 
motion due to gravity in a fluid is 

dv _ _F^ 

dt~ g ~ M' 

where g is the acceleration of gravity and M the mass of the sphere. 

An iron sphere 20 cm. in diameter is released from a height of 1000 meters. How 
much longer will it take to reach the ground than if it were falling in vacuum? The 
following data are in c.g.s. units: 



R 


C W (R) 


R 


C W (R) 


10~ l 


58 


10* 


1.1 


1 


10 


10 6 


1.2 


10 


2.7 


2 X 10 5 


1.15 


10 2 


1.4 


5 X 10 6 


0.3 


10' 


1.0 


10 


0.34 



The fluid is water at 4 C. 



SUMMARY 



187 



4. Solve by graphical or numerical integration the following system of differential 
equations. 



dl 
dt 



(7) 



ds ^^ 
dt ~ 



JW 



The initial conditions are / = 7 0> 5 = So at / = 0. These are the equations of 
dynamic braking of synchronous machines. The armature is short-circuited through 
a resistance r . The dependent variables are the speed s and field current /. The 
constants are : 



x d = 0.64. 
x q - 0.46. 
x' d = 0.29. 



R = 0.002 ohm. 
/o = no load current 



57.5. 



1 1 jump of current = 32. 
r = 0.227. 

P = power = 400 kv-a. 
K = 735.5. 
L = 0.67 henry. 

/ = moment of inertia = 232,000. 
SQ = 94.8 r.p.m. 



First make the following change of dependent variables: 



1 - 1 - 



5. Solve by numerical integration the equation 
NA 



/ NA \ di . _ i(a - be-"') 

\K(A + Bi)* 10 8 / dt + n ~ A -f Bi ' 

This is an equation for the field current * of a shunt excited d-c. generator which 
undergoes an exponential change of speed. The symbols have the following numeri- 
cal values: 



N number of field turns 

K voltage proportionality factor 

r field circuit resistance 



= 4500. 

= 7 X 10~ 8 volt per line per rev. 

per min. 
= 55 ohms. 



188 METHODS OF SOLVING DIFFERENTIAL EQUATIONS 

a final speed = 2400 r.p.m. 

b total change of speed = 1200 r.p.m. 

o reciprocal time constant of speed change = 1 sec." 1 
A constant =15. 

B constant =1.5. 

A and B are constants in the equation used to approximate the magnetization 
curve: 



K(A -f Bif 



CHAPTER III 
VECTOR ANALYSIS 

The chief uses of vector analysis in engineering are: derivation of 
the partial differential equations of mathematical physics; the study 
of vector fields (magnetic, electrostatic, and hydrodynamic) ; and the 
analysis of rotating electrical machines. Since the emphasis of this 
text is on the reduction of physical phenomena to mathematical form, 
we are interested only in those parts of vector analysis which assist in 
these reductions. 

I 
OPERATORS AND LAWS OF VECTOR ANALYSIS 

The application of vector analysis to engineering problems can 
readily be made only after certain notations and laws of manipulation 
are understood. The first section of this chapter is thus necessarily 
concerned with a brief introduction to the purely formal parts of vector 
analysis. Some proofs are left as exercises which appear at the end 
of the section. 

67. Vectors. A vector is a quantity that possesses direction as 
well as magnitude; a scalar is a quantity that possesses magnitude 
only. Quantities such as mass, temperature, and electric charge are 
scalars; velocity, force, and current density are vectors. Vector 
analysis deals with vectors which are defined at a single point as well 
as with the more general case of vectors defined at more than one 
point, as along a line, on a plane, or in a volume. When the vectors 
are defined at a single point, as in the treatment of forces on a rigid 
body, the laws of vector algebra, addition, subtraction, and multipli- 
cation, are applied. For example, the sum of vector forces and reac- 
tions is zero; and if all but one are known, the remaining quantities 
can be determined. Another example of the use of vectors defined 
at one point is the vector treatment of alternating currents. Here the 
length of the vector is proportional to the amplitude of the current or 
voltage and the angular displacement to the relative phase. The vec- 
tors are usually confined to one plane and admit of the usual operations 

189 



190 OPERATORS AND LAWS OF VECTOR ANALYSIS 

of vector algebra. The methods of complex number theory are also 
used to treat the subject of alternating currents. 

In more general vector problems, in which a vector is defined for 
all points in a given region, the principles of vector calculus may be 
applied, as well as the algebra of vectors. Such a region is a vector 
field, for the discussion of which certain theorems are available. 

The subject of vector fields includes gravitational force fields, elec- 
tric and magnetic flux densities, magnetic vector potential, the Poynt- 
ing vector, current density in a solid conductor, temperature gradient, 
and others. The electric and magnetic fields because of their impor- 
tance are chosen for discussion, and other fields are treated by 
analogy. 

68. Nature of Vector Analysis. In 10-20, we have seen how 
the first and second derivatives of the calculus can be combined, by 
means of Newton's laws of motion and Kirchhoff's laws, to describe 
engineering phenomena. Our interest in vector analysis is much the 
same. There exist in vector analysis a number of operators called the 
gradient, divergence, curl, line and surface integrals, etc., which play 
the same r61e in vector analysis that derivatives and integrals play in 
differential equations. These new operators possess physical signifi- 
cance just as do the derivatives. By giving the values of certain of 
these operators throughout a region, a vector field is completely 
described, just as a function in calculus is completely determined 
(except for a constant) for real values of the independent variable, by 
giving its derivative. 

The laws of physics and engineering make possible the combination 
of these operators in equations in much the same way that Newton's 
and Kirchhoff's laws combine derivatives in equations. In the cal- 
culus, the derived equations are ordinary differential equations. In 
vector analysis, the derived equations are partial differential equa- 
tions. After a partial differential equation is obtained, vector analysis 
is, in general, of no further use. The solution of the equation belongs 
to another branch of mathematics. The derivation of the equation 
requires not only a knowledge of vector analysis, but also some knowl- 
edge of physics and engineering. However, it is usually far easier to 
derive a partial differential equation than to solve it. 

69. Algebra of Vectors. Vector algebra is very similar to scalar 
algebra. 

(a) Definitions. Zero and unit vectors are those whose magni- 
tudes are respectively zero and one. Two vectors are equal, if and 
only if, they have the same magnitude and direction. By the negative 
vector A, we mean A with its direction reversed but its magnitude 




ALGEBRA OF VECTORS 191 

unchanged. A vector A may always be considered as -4a, where a is 
a unit vector and A is the magnitude of A. 

(b) Addition and subtraction. C, the sum of A and B, is defined to 
be the vector obtained by placing the initial point of B in coincidence 
with the terminal of A and taking C with its initial point coinciding 
with that of A, and its terminal point with that of B. From Fig. 34, 
evidently A + B = B + A. The sum of three 

vectors E = A + B+D = C + D, where A 
+ B = C. The subtraction of A is defined 
to be the addition of A. 

(c) Vector components. A vector is uniquely 
determined by giving its projections on the 
three coordinate axes. These projections are 

A x = A cos (Ax), A v = A cos (Ay), A g = A cos (Az), where (Ax) 
denotes the angle between A and the positive #-axis. If A + B = C, 
it is apparent geometrically that 

A x + B x = C x , 

Ay ~t~ &y == Cj/f 

A, + B, = C,. 

Let, i, j, and k be unit vectors coinciding with the x, y, and z axes 
respectively. By the definition of addition 

A = A x i + A v ] + A x Vi. 

(d) Scalar and vector products. The scalar product of A by B 
(or B by A) is a scalar defined by the equation A-B = AB cos 6, where 
B is the angle between the positive directions of A and B. The scalar 
product is thus the product of one vector by the projection of the other 
vector upon it. Hence A-B = B-A. Also 

i.i = j.j = k-k = 1, and i-j = j-k = k-i = 0. 

It can be shown that A- (B + C) = A-B +A-C; thus we may write 
A -B = (iA x + JA V + kAJ (LB X + JB V + kB,) 



+ j'iAyB, + j'jAyBy + j ' kA yB , 



X + AyBy + A,B.. 

The vector product of A by B (not B by A) is a vector defined by 
the equation 

A X B = BAB sin 6, 



192 



OPERATORS AND LAWS OF VECTOR ANALYSIS 



where 6 is the angle between the positive directions of A and B and 8 
is a unit vector perpendicular to the plane of A and B. The positive 
direction of A X B is defined to be perpendicular to the plane of A and 
B in the sense of advance of a right-handed screw rotated from the 
first to the second of these vectors through the smaller angle between 
their positive directions. (See Fig. 35.) 

A X B = BAB sin 0. 

Consequently, i X i = j X j = k X k = 0, and i X j = k, j X k = i, 
and k X i = j. Also, A X B = - B X A. It is evident that the 
vector product of A and B may be considered as a vector with a mag- 
nitude equal to the area of the parallelogram having A and B as sides 
and with the direction of the normal to the plane of A and B. 

It can be proved that the distributive law of multiplication, namely 
(A + B) X C = (A X C) + (B X C), holds for vector products (as well 



BXC 




l 




I/ 



FIG. 35. 



FIG. 36. 



as for scalar products). In view of this and the above relations be- 
tween i, j, and k, we may express A X B in terms of its i, j, k compo- 
nents as follows: 

A X B = (iA, + JA V + kA z ) X (IB, + JB V + kB.) 
= i X IA X B X + i X jAiBy + i X VsAJb t 
+j X \A V B X + j X jA,B, + j Xk^yg. 
+ k X L4 A + k Xi4A + k X kA .B. 
= {(AyBT^A.By) + j(A,B x - A X B.) 

+ \n(A x B v - A V B X ). 

The vector product may be written as the determinant 

i J k 



AXB = 



*jL x ** y ** z 

B X By B 9 



LINE AND SURFACE INTEGRALS 193 

(e) Triple scalar product. The product A-(B X C) is a scalar 
called the triple scalar product. Inspection of Fig. 36 shows that it 
is equal to the volume of a parallelepiped with edges A, B, and C. 

Since interchanging the terms in a scalar product does not change 
the sign of the product whereas interchanging the terms in a vector 
product does change the sign of the product, it follows that 

A-(B XC) = (B XC)-A =- (C XB)-A =- A-(C X B). 

Since the volume of the parallelepiped remains the same, no matter 
which face is considered as base, it follows that 

A-(B XC) = (A XB)-C = B-(C X A) = C-(A X B), etc. 

Thus the dot and cross may be interchanged at will and the sign of the 
product remains unchanged so long as the cyclic order of the vectors 
remains the same. The triple scalar product may be written as 



A-(B XC) = (B XC)-A = 



* x ** y ** z 

B x B 

err 

SX ^V 2 



y 



(/) Triple vector product. The product A X (B X C) is defined as 
the triple vector product. The vector product of B X C should be 
formed first, and then the vector product of A with this result. The 
final result may be shown to be 

A X (B X C) = B(A-C) - C(A-B). (See Ex. 3, 76.) 

-/70. Line and Surface Integrals Involving Vectors. Certain defini- 
tions of curl and divergence are based upon the ideas of line and surface 
integrals involving vectors. 

C B 

(a) Line integrals. The integral / F-dr is a line integral. The 

*/A 

vector dr is taken along the tangent to the curve AB as in Fig. 37, 
and the vector F may vary in both magnitude and direction along the 
curve. Alternative forms are 

/B xJB 

F-dr = / FcosBdr, 
JA 

and 

,) (Ux + jdy 



(FJx + FJy -f F4z). 



194 OPERATORS AND LAWS OF VECTOR ANALYSIS 

C B 

If F represents a force on a body, / F -di is the work done by the 

JA 
force as the body moves over the specified path from A to B. 

EXAMPLE 1. To fix the ideas more clearly, let F be the force of 
gravity. Let the curve AB (Fig. 38) be one-quarter of the circum- 
ference of a circle. Let us determine the work W done in moving a 
mass m against the force of gravity from A to B along the curve AB in 
the 2^-plane. There is no friction. Then F = wgk. (The minus 




\. 




FIG. 37. 



FIG. 38. 



sign is due to the fact that the force is in the direction of negative k ) 
Then, 

r = i* + jy + kz = jy + ks, 

dr = jdy + kdz, 
z = r cos 6, 
dz = r sin 6dO, 



and 



W 



= - f F-dr = ~ / ( - mgk)-(jdy + krfs) 
JA JA 

//2 x^r/2 

( wgk)-( r sin 0dd)k = mgr I sin 6dO = mgr. 
JQ 

This is, of course, the work done in raising the mass M a vertical dis- 
tance r. If F varied both in magnitude and direction and C were a 
complicated curve, the integrations would, in general, be more com- 
plicated, but no additional principles would be involved. 

(&) Surface integrals. The integral / / F-ndS = / / F-dS is 
the surface integral of F over the surface 5. Let the curved surface 



LINE AND SURFACE INTEGRALS 195 

of Fig. 39 be divided into infinitesimal rectangles AS. The elements 
so formed may be treated as if plane. A plane surface may be repre- 
sented by a vector whose magnitude is equal to the area of the surface 
and whose direction is that of the positive normal. 

If the elementary plane surface is part of a closed surface, the 
positive direction of the normal is outward. Denote by n the unit 
normal to the surface. Then the vector representing the elementary 
surface in Fig. 39 is dS = nAS. 

The integral / / F ndS has many applications as well as being 

fundamental in definitions. It represents volume rate of flow through 

the surface 5 if F is a velocity 

vector. Consider an incompressible 

liquid flowing through the surface 

of Fig. 39. At the point P(x, y, z) 

let its velocity be F, parallel to the 

2-axis as shown. Since the liquid 

is incompressible, the flow through 

the element of surface AS per unit FIG. 39. 

time will be the same as the flow 

through the element AS* (see Fig. 39) per unit time. But AS = AS 

cos 6, where 6 is the angle between n and the s-axis. Hence, the 

rate of flow is AS,F = AS(cos 0)F 

= ASF-n, 
or the total flow per unit time through the whole surface is 

Flow = / / F-ndS. 




The surface integral of F over a surface is called the flux of F 
through the surface. For example, if heat is flowing through a sur- 
face S, the amount of heat which crosses unit area drawn normally to 
the lines of flow in unit time is called the intensity of heat flow or the 



heat-current density, q, while I / q-dS is the flux of heat. 

EXAMPLE 2. Let the velocity F in Fig. 39 be defined by the 
equation F = 3;yk. Let the surface considered be the octant of a 
sphere, whose equation is x 2 + y 2 + z 2 = R 2 . The flow through the 
surface per unit time then is 





196 



OPERATORS AND LAWS OF VECTOR ANALYSIS 



The spherical surface coordinates R, 0, 0, are related to x, y t z by the 
relations: 

x = R cos <j> cos O t 

y R cos sin 0, 
z = R sin <t>. 
From Fig. 40, dS is seen to be R 2 cos <j>d6d<t>. Hence 



/2 X-7T/2 



Flow 



/T X-7 

/ 
M) 



3R 3 cos 2 sin 6 sin 



Since, in the case of an incompressible fluid, the flow through the 
trace in the :ry-plane is equal to the flow through the spherical surface, 
the above result is easily checked by the integral 

/R x# 

3yxdy = 3(R 2 - y^ydy = R 3 . 
JQ 





FIG. 40. 



FIG. 41. 



71. Vector Operators. Arts. 68-70 are the basis for the definitions 
and interpretations of derivative, gradient, divergence, and curl. 
These concepts, along with relations and theorems involving them, 
make up the nucleus of vector analysis. Many important concepts 
in mathematics have two or more equivalent definitions. That one 
is then employed which is most readily useful in a given situation. 
In 73-75, two definitions of gradient and three each of divergence 
and curl appear. Arts. 72-78 may properly be called the calculus of 
vectors. 

72. Derivatives of Vector Quantities. Let r = ix + jy + kz, 
where x = x(T), y = y(T), z = z (T), and T is any real parameter. 
If the initial point of r is fixed at the origin, the terminal point of r 
varies and describes a space curve as T varies. Let A and B be tw<? 
nearby points on this curve. (See Fig. 41.) Then 



DERIVATIVES OF VECTOR QUANTITIES 197 

Ar = AB TI r (ri is not a unit vector.) 
= i*i + jyi + kzi - ix - jy - ks 
= i(*i - *) + j(yi - y) + k(*i - 2) 
= iA# + jA;y + kA2. 
Dividing Ar and taking the limit 

= i 4. \^L jL.\r^L 
dT dT^^dT^ dT' 

It is evident that, as B approaches A (Fig. 41), the vector repre- 
senting Ar approaches the position of the tangent to the curve at A. 

dr 

Hence, is a vector tangent to the space curve described by the 
dJ. 

terminus of r. Thus, it follows that the derivative of a vector having 
constant magnitude but variable direction is a vector perpendicular 
to the differentiated vector. 

By treatment similar to the above, 

\ 

dT 2 ' 

Formulas for differentiating P-Q and P X Q may be obtained by 
expressing each product in its expanded form and taking derivatives 
of these forms. Thus, 

dr (p ' Q) = 7r Qt + 7r Qv + ~dT Q ' + Pz dT + p "dr + Pt dT 



= dT' Q+P 'dT' 



and 



Both products are differentiated by differentiating the factors just as 
in the case of algebraic products, paying no attention to the dot or 
cross. It is important to notice, however, that in taking the derivative 



198 OPERATORS AND LAWS OF VECTOR ANALYSIS 

of the vector product the order of the vectors must not be changed 
unless the sign is changed. 

73. Gradient. Let V(x, y, z) be a scalar point function. Suppose, 
for example, V is the temperature at any interior point P(x, y, z) of a 
body. In general the scalar V will have different values at neighbor- 
ing points; and the rate of change of V with respect to distance will 
depend upon the direction in which the distance is measured. Two 
useful and equivalent definitions for the gradient of a scalar are the 
following: 

1. The gradient of Fat point P is the vector having the direction 
of the greatest rate of increase of V, with respect to distance, at P, 
and a magnitude equal to this rate of increase. The symbol for the 
gradient of V is V V. 

2. The gradient of V is also defined with respect to cartesian coordi- 
nates 

. 9F , .9F , , QV 
VV = 1 -- h J - + k . 
dx dy dz 

By this definition the symbol V, called " del," is equivalent to a vector 
operator, 

r7 . 9 . . 9 , , 9 

V s i + j - + k . 

dx dy dz 

Two definitions are said to be equivalent if each implies the other. 

The equivalence of these two definitions is 
shown as follows. Consider the neighboring 
surfaces V(x, y, z) = constant C passing 
through P, and V(x, y, z) = C + dC. (Fig. 
42.) The normal derivative of F, written 
d V/dn, is the rate of change of F along the 
FlG 42 normal to the surface. The rate of change 

of F in any other direction dr, dV/dr, is 

less than dV/dn because dV = dC is constant on passing from C to 
C + dC and dn < dr. Thus, by definition 1 the gradient of F 
always has the direction of the normal to the surface V(x, y, z) = 
constant, or 






dn 

where n is a unit vector in the direction of the normal. From the last 
equation, the derivative of F in a general direction r is easily found. 



GRADIENT 199 

Forming the scalar products of each side of the last equation with the 
vector di we obtain 

dV 

dr-VV = n-dr - 
dn 

dV 

= dr cos 6 -7 = dV, 
dn 



or dividing by dr 

dV dr-VV 
dr ~ dr 



(242) 



where ri is a unit vector parallel to dr. The last equation is important 
since it shows that the derivative of V in the direction di is the pro- 
jection of the gradient in that direction. This relation gives the com- 
ponent of V V in the x direction by replacing ri by i and r by x. Thus 



2T-I.VI, 



Similarly, 



Combining gives 

VF = i hj h k 

which is the second definition of gradient. V V is frequently written 



The following physical examples serve to explain some applications 
of the gradient of a scalar. Consider the flow of heat through a solid 
body, and assume that it is possible to select a curvilinear section 
through the body at all points of which the temperature is the same. 
Call this section a level surface or an equipotential surface. Let V 
be the temperature of the body at any point P(x, y, 2), and let q be 
the vector representing the direction and intensity (flow per unit area) 
of heat flow at the point P. It is known from the theory of thermal 
conduction that the flow of heat will be in the direction of the greatest 
decrease of temperature and will have a magnitude per unit area pro- 



200 OPERATORS AND LAWS OF VECTOR ANALYSIS 

portional to this rate of change of temperature. But V V is a vector 
with the magnitude of the greatest rate of increase of temperature 
and in the direction of this greatest rate of increase. Therefore, 
changing the sign and employing the proportionality constant k we 
obtain 

q =_ kVV. 

This is known as Fourier's law of heat flow. The constant k is the 
thermal conductivity of the material. When V is known as a function 
of x, y, and z, the vector representing the intensity of the heat flow is 
readily found from the above equation. 

For a second example of gradient, consider a cloud of water vapor 
tyhose density varies from point to point. Suppose that the density 
increases toward the geometrical center and is maximum at that point. 
Points at which the density has some fixed value may be selected within 



Source 
Of heat 





FIG. 43. FIG. 44. 



the cloud. The curvilinear surface containing all those points is a 
level surface or an equipotential surface for that particular density. 
A line drawn from a point on this surface in the direction of the greatest 
rate of increase of density with a length equal to this rate of increase 
will coincide with the vector representing the gradient at that point. 
Inspection of the figure shows that, in general, the gradient will vary 
in direction and magnitude from point to point along the same equi- 
potential surface. It is not necessarily directed toward the point of 
maximum density. 

74. Divergence. Three equivalent and useful definitions of the 
divergence of a vector function F(#, y, z), are the following: 

1. The divergence of F(#, y, 2), denoted by V -F, is defined by the 
equation 

v . p= 9- + 9 + 9 
dx 'dy Qz ' 

2. The divergence of J?(x, y, z) is defined also by the equation 

,. F = 1 . +) . +k .. (243) 



DIVERGENCE 



201 



These two definitions are especially useful in establishing operator 
formulas. The third definition, now given, is especially useful in 
physical applications in connection with vector fields. 

3. The divergence of F(#, y, z) is defined by the equation 



(244) 



V 



where n is the exterior normal to the closed surface S whose volume 
is V. Integrals of the form / I n-FdS have been discussed in 70, 

Thus the divergence of the vector F at a point is the net outward 
flux of F per unit of volume as the volume, which includes the point, is 
made infinitesimally small. 

The equivalence of the first two definitions is established as follows: 

?>F aF aF 

v .F = i- +J- + k> 

'doc Qy ] 'dz 



93 



33 



i 



+ 



(245) 



Also, beginning with the last equation, we may obtain the first. 
Hence, since each definition implies the other, the two are equivalent. 
The equivalence of the third definition with 

* 

either of the first two will be shown later in 
this chapter after Gauss's theorem has been 
proved (Ex. 1, 87). 

For a physical application of divergence, 
consider a small parallelepiped of dimensions 
dx, dy, dz in a mass of liquid. Assume that 
liquid may be flowing through all six faces 
of the parallelepiped. Let M represent the mass of the liquid flow- 
ing through a unit cross-section in unit time. (The direction of M 
is that of the velocity of flow.) Let p represent the density, and V 




FIG. 45. 



202 OPERATORS AND LAWS OF VECTOR ANALYSIS 

the velocity ( V x in a direction parallel to the x-axis, V v in a direction 
parallel to the y-axis, etc.); then the mass flowing through a unit 
area in unit time will be the density times the velocity normal to 
that area, or 

M = P V. 

The flow to the right per unit time through the face ADEH is then 

Mydxdz = pVydxdz, 
and the flow to the right per unit time through the face BCFG is 



M V + dy dxdz 

dy 



= \ P V V + 2> dy] dxdz. 
L dy J 



The net increase of fluid in the parallelepiped (due to the flow through 
the two faces only) is then 

Mydxdz My + - v dy dxdz = --- v dxdydz. 
L dy J dy 

Treating the flow through the other faces similarly and adding the 
results, we have, for the total increase in mass inside the parallelepiped 

in unit time, / . f \ 

I dM x dM v dMA 
i ------ i dxdydz, 

\ dx dy d* / 

or dividing by dxdydz we obtain for the total increase in mass per unit 
time per unit volume 

9M, 



9* dy 92 
This is recognized as V -M. Thus, 



x y 

--- 



dx dy 

And since the total increase in mass per unit time inside the parallele- 
piped is also dxdydz, 



This last equation is called the equation of continuity, and is an 
expression of the principle of conservation of matter. When the 
liquid is incompressible, it is evident that 

p = constant and V M = 0. 



CURL 



203 



The divergence is the excess of the outward over the inward flow, and 
the convergence is the excess of the inward over the outward flow, 
both per unit of volume. When the divergence of a vector function 
of position in space vanishes in a region, the function is said to be 
solenoidal in that region. 

75. Curl. Three equivalent and useful definitions of the curl of a 
vector function F(x, y, z) are the following: 

1. The curl of F(#, y, z), denoted by V X F, is defined by the 
equation 



dy 



J 



333 
3* dy dz 

F F F 
r x fy FZ 

2. The curl of F(#, y, z) is defined also by the equation 

^\TJ* oT? c^T? 

. O" . O* O' 

~~ dx dy 82* 



(246) 



(247) 



The equivalence of these two definitions is shown by replacing the 
dots by crosses in Eq. (245). 

3. The final definition is very important in physical applications. 
The curl of F is a vector and hence has components in any direction. 
To find the component in any direction s at any 
point P choose the direction and describe a small 
circular area normal to the direction at the point in 
question. (See Fig. 46.) Form the line integral of p IG ^ 
the vector field in the conventional positive direction 
around the circle. (The positive direction of traversing the boundary 
of an area is related to the positive normal as the direction of rota- 
tion of a. right-handed screw is related to its direction of advance.) 
Then the quotient of this line integral by the little area is, in the 
limit, the $ component of the curl, or 



(CurlF), s 



/F-dr 



a-0 



(248) 



If three components of the curl are determined, the curl may be found 
by adding them vectorially. 

The equivalence of this definition and those preceding is shown 
later. (Ex. 2 87). 



204 



OPERATORS AND LAWS OF VECTOR ANALYSIS 



Physically the curl may measure the tendency of the vectors of 
the field to run in closed loops. That is, the curl of the electric field 
about a point charge is zero, as may be seen by taking line integrals. 
But the curl of the linear velocity field of a rotating body is twice the 
angular velocity, and directed parallel to the axis of rotation. If the 
curl of a vector function of position in space vanishes everywhere in 
a region, the function is said to be irrotational in the region. The 
electrostatic field, the gravitational field about attracting matter, and 
the field of magnetic intensity in a region containing no current are 
irrotational vector fields. The general vector field has a curl and 
divergence neither of which is zero. 

76. Operator Formulas. There exist a number of formulas involv- 
ing vector operators. These arc used in making transformations in 
vector fields and in deriving partial differential equations. The fol- 
lowing are eight important ones. Of these, two are proved. The 
remainder can be proved in a similar manner. The symbols <t> and 
F represent any scalar point function and any vector point function, 
respectively. A and B also are general vector point functions. 



3. V X V<f> = 0. 

4. V (V X F) = 0. 

5. V X (V X F) = V(V-F) - V-VF. 

6. V-(M) = 4>V-A + A-V</>. 

7. V X (A X B) = AV-B - BV-A + B-VA - A-VB. 

8. V(A-B) = A-VB+B-VA+AX(VXB) + B X (V X A). 
EXAMPLE 1. Prove formula 3. 



VX V<t> = 



a* 



j 



doc dy 



k 

9 



\dydz 



\dzdx 



dxdy 



OPERATOR FORMULAS 
EXAMPLE 2. Prove formula 5. 



205 



X (V X F) = 



j 

A 

dy 



k 



dy 



Qz 



g* dy 



= i [A (tt _ 9^A __ A f 9*f _ 

LaAa* dy/ 82X92 



9* 



[JL 

L92\ 
[A 

Lg* 



92 



dy 



k _ 



= 



etc. 



= V(V-F) - V-VF. 



EXERCISES 



1. Compute by vector methods the area of the triangle whose vertices are (7, 3, 4), 
(1,0, 6), and (4,5, -2). 

2. Compute both the scalar and vector products of the pairs of vectors 

A = 0.6i + 4j - 3k, 
B = 4i + Oj + 10k, 
C = 0.7i + 9j -k, 
D = i + 3j - 7k. 



3. Let A = aii + a 2 j + flak, B = M -f 6 2 j + 6 3 k, and C = Cii -f 
form the expansions A-C, A-B, and A X (B X C), and show that 



+ 



. Per 



also 



A X (B X C) = B(A-C) - C(A-B), 
(AXB) XC = (C-A)B- (C-B)A. 

r = ae r + be' 7 *, 
where a and b are constant vectors, show that 



4. Given that 



d*r r iv -r 

= ae +be 



f , 



206 MATHEMATICAL PHYSICS OR VECTOR FIELDS 

5. Find the derivative of the scalar point function 

1 

in the direction of the vector ix -f- jy -f kz. 

6. If r (x 2 + y 2 + z*-)M, show that V-V(l/r) = 0. 

7. If 



r i* + jy + to (r = V x 2 -f- y 2 -|- s 2 ) 

show that 



8. Prove the six unproved formulas of 76. 



n 

DERIVATION OF THE PARTIAL DIFFERENTIAL EQUATIONS OF 
MATHEMATICAL PHYSICS OR VECTOR FIELDS 

The derivation of the partial differential equations of mathematical 
physics is little more than expressing vector relations which hold within 
a vector field or between vector fields. The basic relations themselves 
are, in general, physical relations in vector form accompanied by cer- 
tain mathematical transformations resulting in the partial differential 
equations of 79. 

77. Some Vector Fields. There are many kinds of vector fields. 
In the study of heat conduction, it is known that the flow of heat is in 
the direction of the greatest decrease of temperature and has a mag- 
nitude per unit area proportional to the rate of change of temperature. 
This statement is expressed simply by the equation q = kV V, where 
q is the heat flowing through a cross-section of unit area per unit time, 
the direction being that to give maximum q; F is the temperature, a 
scalar function ; and k is the thermal conductivity of the body. Near 
every mass there is a field of force called the gravitational attraction. 
This force of attraction at any point may be obtained by taking the 
gradient of a scalar point function called the gravitational potential 
(see 82). Likewise, near an electrically charged body, there is the 
electrostatic field. 

At points exterior to the charge, there exists a scalar point function, 
the electrostatic potential, whose gradient taken at the point P(x,y,z) 
gives the negative of the electric intensity at that point. Near a mag- 
netized body there is a magnetic field. The negative of the magnetic 



PRELIMINARY THEOREMS 207 

intensity of this field is given by the gradient of the scalar magnetic 
potential. Within a body of flowing fluid there is a vector field or 
velocity field. If the curl of this field is zero then there exists a func- 
tion <, called the velocity potential, such that the gradient of <t> at 
any point gives the negative of the velocity of the fluid at that point. 
In 10 the forces acting on the body may be viewed as a limiting case 
of a vector field and Eq. (10) may be written imx" + ikdx' + ikx = 0, 
where i is the unit vector directed along the #-axis. 

78. Preliminary Theorems. Before deriving the partial differen- 
tial equations of mathematical physics, it is necessary to understand 
the very important theorems, in vector analysis, of Gauss, Stokes, and 
Green. In 70, line and surface integrals involving vectors were 
defined and illustrated. The concept of the volume integral, 

/ Fdv, of a vector function is also needed. The integral / Fdv is 

J J V<A 

defined by the equation 

f Fdv = i fpjv + j jFJv + k JP4v. (249) 

Vol Vol Vol Vol 

The three theorems of this section are the machinery by which trans- 
formations are made between line, surface, and volume integrals. 
Eqs. (250-252) state in symbols, respectively, Gauss's, Stokes's, and 
Green's theorems as follows: 

/ It V-Fdv = JJF-dS t (250) 

Vol S 

J J V X F-rfS = /Vdr f (251) 

(252) 



Vol 



Gauss's theorem stated in words is: The volume integral of the 
divergence of a vector function of position in space taken over a volume 
is equal to the surface integral of the vector function taken over a 
closed surface bounding the volume. To illustrate Gauss's theorem 
qualitatively, consider a mass of metal within which heat is generated, 
say by electric current. Gauss's theorem states that the total heat 
flowing, in the steady state, out through the surface is equal to the 



208 MATHEMATICAL PHYSICS OR VECTOR FIELDS 

volume integral of the divergence of the heat-flow vector, which can 
be shown to be equal to the amount of heat generated in the solid. 

Stokes's theorem is: The surface integral of the curl of a vector 
function of position in space taken over a surface S is equal to the line 
integral of the vector function taken around the periphery of the sur- 
face. A physical illustration of Stokes's theorem may be had in the 
magnetic field about a wire carrying a current. According to the 
circuital theorem the work done in carrying a unit pole around a closed 
path is 4rr times the current enclosed by the path, or if the path lies 
in air (/x = 1), in symbols 

r 

E-dr = 47r7. 

But / is equal to the surface integral of the current density j over any 
surface bounded by the closed path c, 



= 47r/ J 



j-dS. 

The circuital theorem may also be written V X B = 4?rj. To see that 
this is true it is only necessary to refer to the third definition of curl 

in 75. If in Eq. (248), F is replaced by B and / E-dr by 4?r/ we 

*/c 

obtain 



/ E dr 

|. /! 



47T/ 

(V X B)y = lim = lim = 4arj. 

a-K) a a->0 a 

Replacing in the double integral above, 4rrj by V X B we obtain 
J E-dr =J j V XB-dS, 

which is Stokes's theorem. The material of this article is a statement 
and illustration of these theorems by means of physical examples. 
But these theorems depend in no way upon physical experiment. 
They are mathematical identities. 

79. The Partial Differential Equations of Mathematical Physics. 
The chief partial differential equations of Mathematical Physics are 
the following: 

a. Laplace's equation V V V = 0, which is satisfied by the functions: 
1. Gravitational potential in regions unoccupied by attracting 

matter. 



EQUATION OF HEAT CONDUCTION 209 

2. Electrostatic potential at points where no charge is present. 

3. Magnetic potential in regions free from magnetic charges. 

4. Temperature in steady state. 

5. Velocity potential at points of a homogeneous non-viscous 
fluid moving irrotationally. 

6. Electric potential in homogeneous conductors in which a cur- 
rent is flowing. 

b. Poisson's equation V-VF = e. 

c. Equation of heat conduction without sources, a 2 V-V0 = Of 

d. Equation of heat conduction with sources, a 2 (V-V0 + e) = 6 t . 

e. Wave equation, a 2 V -V\l/ = ^*, and 
/. a 2 (V-V^ + e) = *. 

g. Equations of elasticity. 

h. Telegraphists' equation, a< + b<t> t + cV-V<t> = ce. 

i. Maxwell's field equations. 

j. Euler's equation for the motion of a fluid. 

The single subscript / indicates one partial differentiation with 
respect to time; two subscripts, partial differentiation twice. We now 
derive, in vector notation, some of the above equations. 

80. Equation of Heat Conduction without Sources, a 2 V-V0 = 6 t . 
Consider the following problem: A mass of iron has been heated to a 
certain temperature and left to cool. What is the temperature at any 
point of the mass at any time ft The differential equation giving this 
temperature may be found from the following physical facts: 

(a) The flow of heat will be in the direc- 
tion of the greatest decrease of tempera- 
ture and will have a magnitude per unit 
area proportional to this rate of change of 
temperature. 

(b) The rate at which heat is lost by a ^ElemenToflsothermal surface 

given region of the body is the heat flux pass- FIG. 47. 

ing through the surface bounding the region. 

The rate of heat loss from an element of volume dv in terms of 

P\o 

temperature 6 and specific heat c is cp dv where p is the density. 

3^ 

Thus the rate of heat loss from a general region of volume V (See 
Fig. 47) bounded by surface 5 is 




9<? r r r w , 

= I / I Cp dv. 

Qt J J V J Qt 



210 MATHEMATICAL PHYSICS OR VECTOR FIELDS 

In general S is not an isothermal surface. We may also express the 
rate of heat loss in terms of the heat current density q (heat flow per 
unit of time per unit area normal to the flow) as 



--// 



Equating these two expressions by relation (6) 

- f f f "%*" fj *"**- (253) 

By means of Gauss's theorem, the last equation becomes 




Since these integrals are equal for every volume, the integrands must 
be equal. Hence 

90 rr 

- pC _=V.q. 

But by relation (a), q = kVO, where k is the thermal conductivity. 
The last equation then becomes 



70 = 0| f 

where a 2 = . 
P 

81. Equation of Heat Conduction with Sources. In this case, 
physical relations (a) and (6), 80, still obtain, and also one additional 
one. Each element of the mass within the volume V may have heat 
generated in it by some means, for example, by an electric current. 
The density of strength of source e of heat is defined by the equation 

,. Total heat created within V per unit time 

e = hm ; . 

V~ V 

The additional physical relation is: the rate at which heat is emitted 
from the element of volume dv may be considered as consisting of two 
parts : first, that which is the rate of cooling the element if no source 

>-\Q 

were present, namely, - pc dv; and secondly f that du to the 



CONCEPT OF POTENTIAL AND THEOREMS 211 

source edv. Returning to Eq. (253) of the preceding paragraph, we 
write 




Vol 

Since this equation holds for every volume, it follows that 

a 2 (V-V0 + e) = fli. 

82. Concept of Potential and Theorems of General Vector Fields. 
It has been noted in 77 that the gradient of a scalar point function 
(called various kinds of potential) gives a vector field. This leads to 
the definition of a potential. A potential is a scalar point function 
whose gradient is a vector field. In such a case, the vector field is 
said to possess a potential. It is by no means true that all vector 
fields possess a potential. The simple criterion for the existence of a 
potential is given by the theorem: 

I. A necessary and sufficient condition that a field F possess a 
potential is that V X F = 0. (See 27 for the meaning of necessary 
and sufficient.) 

To determine whether the curl of a field is zero, it is necessary to 
know physical facts about the field and then to apply Eq. (248). For 

instance, in the magnetostatic case, if the line integral / B-dr is cal- 

JQ 

culated around a closed path which encloses no currents, by the cir- 
cuital theorem, / B-dr = 0, and consequently, by (248), curl B = 

in such regions. Similarly, the line integrals of the force of attraction 
and electric intensity, taken around closed paths, are zero for gravi- 
tational and electrostatic fields. 

The concept of potential function is one of the most important in 
mathematical physics because, once the potential (if it exists) of the 
field is known, the field is determined. This raises the question, why 
not find the field due to the distribution of charge, current, or mass 
at once, and dispense with the intermediate potential? The answer 
is that the potential satisfies certain partial differential equations which 
can be integrated and hence the potential may be found with less 
difficulty than the field. The following table displays some of the 
important potentials and their definitions, 



212 



MATHEMATICAL PHYSICS OR VECTOR FIELDS 





Definition by line 
integral 


Definition by vol- 
ume integral 


Definition by partial 
differential equa- 
tion. Solution, sub- 
ject to boundary 
ary conditions of: 


Newtonian potential 


Negative work 
= / r F.(+dr) 

CO 

per unit mass 


v-C?*L 

J r 


V-V7 = 0, 

or V V V = 47TM. 


Electrostatic poten- 
tial 


Work 
-jfB.(-*) 


v = f>^ 

J Y 


V-VK = 0, 
or V V V 4?rp. 




per unit charge 






Magnetic potential. . 


Work 
-jTW-*) 

per unit pole 


n-Tf* 

J r 


v-va = o, 

or V-Vft = 47r<r 


Magnetic vector 




A-/"** 


V-VA = 0, 


potential 




J r 


or V V A = 4?rj 










Velocity potential. 






v-v# = o 










Velocity vector po- 




$-L r dv 


V-V4 = 0, 


tential 




V 9^7 r ay 


or V- V< = 2w 






^TT VO I r 





In the above table: 

\i = mass per unit volume, 
p = density of charge per unit volume, 
<r = pole strength per unit volume, 
* = velocity vector potential, 

o> = angular velocity of fluid = ^ curl of linear velocity, 
F = gravitational force, 

H = magnetic intensity (force per unit pole) = B/n, 
<|> = velocity potential, 
E = electric intensity. 

In the case of vector potentials the fields desired are obtained not 
by taking the gradient but by taking the curl of the vector potential. 

From theorem I, it is evident that vector fields possessing potentials 
are not the most general fields since the curls of such fields have the 
special value zero. What then is the nature of a general vector field, 



EQUATIONS OF GRAVITATIONAL, ELECTROSTATIC FIELDS 213 

and what must be known about a general field to determine it? The 
answer to these two questions are theorems II and III. 

II. Let F be a single-valued vector function which, along with 
its derivatives, is finite and continuous and vanishes at infinity. 
Then F can be written 

F = V<t> + V X H, 

where <t> and H are respectively a scalar and a vector point function. 
This is the Helmholtz theorem in vector analysis. 

III. A vector field is uniquely determined if the divergence and 
curl be specified, and if the normal component of the field be known 
over a closed surface, or if the vector vanish as 1/r 2 at infinity. If 
neither of the last two conditions is satisfied, the field is determined 
except for an additive constant vector. 

We now resume the derivation of equations. 

83. Partial Differential Equations of Gravitational, Electrostatic, 
and Magnetostatic Fields. These derivations are based upon Gauss's 
law and in the case of the magnetostatic field, the circuital theorem 
of Ampere. 

1. Gauss's law. In electrostatics the force between two charges q\ 
and 32 is given by the inverse square law 



The field vector E is defined as the force per unit charge. Gauss's law 
relates the surface integral of E over a closed surface 5 to the charge Q 
within S. For a region containing no polarized dielectric it is 

E-dS = 47r<2. 

This can readily be proved from the inverse square law; in fact, it is 
a mathematical equivalent which is based on no further experimental 
evidence. Thus if a phenomenon is characterized by Coulomb's 
inverse square law, as the magnetostatic and gravitational fields are, 
Gauss's law also holds. For the magnetostatic field E in Gauss's law 
is replaced by H, the force per unit pole, and Q, by the number of unit 
poles enclosed. For the gravitational field, E is replaced by F, the 
force per unit mass, and Q by M, the negative of the total mass 
enclosed. The negative sign occurs because the force between masses 
is attraction whereas that between like charges or like poles is repulsion. 



2. The circuital theorem. The line integral / H-dr of the mag- 



214 MATHEMATICAL PHYSICS OR VECTOR FIELDS 

netic intensity H, due to a current, taken around any closed path c 
encircling a conductor is equal to 47r/, where / is the total current 
flowing in the conductor. 

By means of 1 and 2 above, most of the fundamental laws 
governing gravitational, electrostatic, and magnetostatic fields are 
quickly obtained. 

(a) Gravitation. In gravitational fields Gauss's law is 

F-dS = 4?rlf, 

where M is the total mass enclosed. The last equation may be 
written 



/ / F-</S =- 47r I idv, 
JsJ fa 



where /x is the mass density. Applying Gauss's theorem (250), we 
have 



V-Fdv =- 4?r / ydv. 

Vol 



Since the last equation holds for every volume, it follows that the 
integrands are equal, that is, 

V-F=-47TM. (254) 

By applying the definition of curl (248) to a gravitational field, which 
obeys the inverse square law, it can be shown that V X F every- 
where. By theorem I, 82, a potential V exists such that W = F. 
Hence (254) can be written 

WF=-47r/x. (255) 

This is Poisson's equation. It holds at all points occupied by matter. 
At points free from attracting matter p. = 0, and Poisson's equation 
becomes Laplace's equation 

V-VF=0. (256) 

Eqs. (255) and (256) are the important partial differential equations 
of gravitational theory. 

(b) Magnetostatics. Replacing E by H and Q by / <r dv in Gauss's 

Vol 

law, and repeating the reasoning immediately preceding (254), we have 

V-H = 47TO-. (257) 



EQUATIONS OF GRAVITATIONAL, ELECTROSTATIC FIELDS 215 

The quantity <r is the pole strength per unit volume. By the circuital 
theorem and the definition of curl (248), it follows that in non-current- 
carrying regions 

V X H = 0. 

Hence by theorem I, 82, potential function 2 exists in non-current- 
carrying regions such that V12 = H. Hence (257) becomes 



(258) 
At points devoid of magnetic poles the last equation becomes 

V-Vfl = 0. (259) 

In current-carrying regions, by the circuital theorem, V X H 5^ 0, and 
consequently no scalar potential 12 exists. 

(c) Electrostatics. By retracing the steps employed in (a) of this 
article, it follows that 

V-E = 4?rp, 



V X E = 0, 
VV =- E, 

V-VF =- 47rp. 



(260) 



The quantities p and V are defined in 82. So far, the electrostatic 
charges considered in the application of Gauss's law have been free 
charges. Gauss's law as stated above holds only if there is no dielec- 
tric medium within the closed surface. Suppose now, in addition to 
free charges, there is within 5 a dielectric containing bound charges 
which are influenced by an electric field. The field causes the positive 
atom cores and negative electrons of an atom to be displaced from their 
equilibrium (normal) position. The result is that the atom forms a 
dipole. The product of either charge of a dipole by the separating 
distance is called the magnitude of the electric moment of the dipole. 
If the direction is taken from the negative charge to the positive charge, 
the product of this unit vector by the magnitude of the moment is 
called the electric moment, a vector quantity. The polarization P 
of a dielectric is defined to be the total electric moment per unit volume. 
It can be shown that the polarization of the atoms of dielectric is 
equivalent to a mean charge per unit volume of V-P. Hence 
Gauss's theorem becomes 



r , c x 

I E-dS = 4?r(<2 - / V-P<fo) 
/s ^v 



216 MATHEMATICAL PHYSICS OR VECTOR FIELDS 



or 

(261) 

The quantity E + 4?rP is called the electric displacement and is 
denoted by D. Hence Gauss's law for all charges within S is 



/ 
/s 



Again proceeding as in (a), we have 

V-D = 47rp (262) 

instead of 

V-E = 4?rp. 

We are now in a position to derive Maxwell's field equations. 

84. Maxwell's Equations. For the derivation of these equations, 
in general form, there are needed: (a) certain results of 83, (b) the 
experimental results of Faraday and Ampere, (c) vector relations, and 
(d) Maxwell's generalization. 

(a) Results of 83. In Eqs. (257-262) electrostatic and electro- 
magnetic units are employed. The most important of these equations, 
if written in Heaviside-Lorentz rational units (to eliminate the 47r's), 
are 

V-D = p, 

V-B - 0, 

UF I (263) 

V-fc = p, 


H-dr = I. 

(I is in rational electromagnetic units.) 

Eqs. (263) hold for steady currents and stationary electrostatic 
charges and stationary circuits. It is natural to expect the existence 
of a set of simultaneous partial differential equations describing the 
more general electromagnetic configurations, that is, those configura- 
tions or systems in which there are moving circuits and charges not at 
rest. These equations are the well-known field equations. 

(b) Experiments of Faraday and Ampere. In 1831 Faraday dis- 
covered the fact that, whenever the magnetic flux through a closed 
single-turn circuit varies, there is induced in the circuit an electro- 



MAXWELL'S EQUATIONS 217 

motive force whose magnitude is equal to the time rate of decrease of 
flux. The direction of the electromotive force is related to the direc- 
tion of flux through the circuit as shown in Fig. . nux 
48. If the electromotive force is induced in a 
conductor a current flows. 

Ampere first obtained experimentally the re- s^~~~ 
suits upon which the theorem stated in 83 is 
based. F 

(c) Mathematical expression of Faraday's and 

Ampere's laws. The electromotive force e around the closed curve 
C formed by a circuit is defined by the line integral 

e = / E-dr, 

/s 

taken around the curve. By Stokes's theorem, 

/ E-dr = I I V X E-dS, (264) 

where S is a cap (surface) whose periphery is the circuit or curve C. 
Faraday's experimental result, expressed in vector form, is 



or 



f Av X E) -dS = - ~ c r A -dS, 

where the dot over a quantity indicates partial time differentiation, and 
c is a constant of proportionality, equal to the velocity of light, neces- 
sary in this system of units. Since the last equation is true for every 
surface S, it follows that 

V XE =--B. (265) 



c 



Eq. (265) is Faraday's law in differential form. Ampere's circuital 
theorem, in vector notation, is 



/' -//'* 



/c 

where 5 is a cap whose periphery is C. By Stokes's theorem, we also 
have 



/H-dr =JjV X H-dS. 



218 



MATHEMATICAL PHYSICS OR VECTOR FIELDS 



Consequently, 



// 




v X H. - 



or by the reasoning preceding Eq. (265) 

V X H = j. 



(266) 



Eq. (266) is Ampere's law in differential form. If it is assumed that 
Gauss's law is valid for variable fields as well as for electrostatic and 
magnetostatic fields, we then have the four equations: 

V D = p, 

V-B == 0, 



V XE =--B, 
c 



V X H = j = 



(267) 



where v is the drift velocity of charge of density p. 

(d) Maxwell's generalization. Maxwell noted that Eqs. (267) are 
inconsistent with the equation of continuity of charge. The equation 

3p 
of continuity of mass, = V-M, was derived in 74. If p denotes 

3* 

charge per unit volume and v its velocity, the equation of continuity, 
in electromagnetic theory, becomes 

^=-V-(pv). (268) 

ot 

This equation merely states that the time rate of increase of charge 
in any region is equal to the excess of charge flowing in, per unit time, 
over that flowing out. All experimental evidence indicates that the 
law of continuity holds, that electricity is neither created nor 
destroyed. 

The contradiction between Eq. (268) and the first and last of (267) 

pv 

is seen as follows. Taking the divergence of V X H = , we have 

c 



= V-V XH = 0, 



(269) 



or 



V- = 



MAXWELL'S EQUATIONS 



219 



But (268) gives 



/ N 

V-(pv) = . 



Moreover, if the first of (267) is differentiated with respect to time, 
there is 

2-'* 

and from the value of in the equation of continuity 

V-(pv) =- V-D. (270) 

Equations (269) and (270) do not agree. Accordingly, Maxwell 
revised Ampere's law as follows. Let the total current consist of a 

pv I) 

convection current and a displacement current . The j in equa- 
c c 

tion (266) is then replaced by - (pv + D), and Ampere's equation as 

c 

revised by Maxwell becomes 



V X H = - (pv + D). 

(s 



(271) 



If the divergence of (271) is taken, Eq. (270) is obtained. Bui 
(270) is a consequence of (268) and the first of (267). Thus the equa- 
tion of continuity is satisfied if system (267) be replaced by the equa- 
tions 

V-D = p, 



V-B = 0, 
V XE = --B, 



V X 



(pv 



(272) 



These are the field equations of Maxwell. 

If the currents are steady the D = and (272), in this special case, 
reduce to (267). 

In regions devoid of charge and current equations (272), since 
B = /iH and D = E. become 



220 



MATHEMATICAL PHYSICS OR VECTOR FIELDS 



V-E = 0, 
V-H = 0, 

V XE =- 



VXH = -E. 

c 



(273) 



The constants ^ and k are respectively the permeability and dielectric 
constant of the space for which (273) are valid. 

The nature of the solution of these equations is discussed in 86, 
and the equations are solved in Vol. II, Chap. Ill, for configurations 
of charge and current of great industrial importance. 

85. Euler's Equation for the Motion of a Fluid. The physical 
principle on which Euler's equation for the motion of a perfect fluid 
rests is nothing more or less than Newton 's law of motions explained 
in 9. But since we shall need to consider the acceleration of a mass 
of fluid, it is first necessary to derive the mathematical expression for 
this acceleration. 

A perfect fluid is one which cannot support a tangential stress. 
Let v be the linear velocity of an element of fluid at the point 
P(x, y y z) at the time /, and v + dv its velocity at Q(x, y, z) near P at 




FIG. 50. 



the time / + dt. Obviously, v is a function of the four variables 
x y y, *, and /. Hence, by the expression for a total differential, 



dt 
where 



_ 9v dy_ 3v dz Qv 

~ 



, (274) 



. , . 

W = 1 h J h k . 

9* dy 32 



dv 



Evidently is the acceleration of the fluid at P. 
dt 



SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 221 

Let the mass of fluid in an infinitesimal rectangular parallelepiped 
dxdydz be pdxdydz. Suppose that this infinitesimal mass is acted upon 
by a field of force F per unit mass of the fluid, and also by the pressure 
p due to the remainder of the fluid. The pressure is normal to the 
faces of the parallelepiped. Let the force due to the pressure p acting 

on the face A be ipdydz. Then the force on B is i ( p H -- dx J dydz. 

\ O"' / 

The net force acting to the right due to the pressure is i dxdydz. 

(jX 

Let the ^-component of the force F be F x . The total force acting 

to the right then is i(pF x -- - ) dxdydz. By Eq. (274), the accel- 
\ 3#/ 

eration of the infinitesimal mass dxdydz is ( -- (-v-Vv). Denote 

\3/ / 

the ^-component of this acceleration by ( + v-Vv) . Since the 

\3^ / z 

sum of the components of all the forces acting, including the inertial 
reaction, is zero, we have 



+ v 



-Vv) pdxdydz = i( P F x - jdxdydz. (275) 

/ x \ Qx/ 



We have two similar equations for the components of the forces along 
the y- and z-axes. Adding these two equations to Eq. (275), there 
results, 

<^ + vVv = F --Vp. (276) 

3* p 

This is Euler's equation of fluid motion. If F is derivable from a 
potential ft, (276) becomes 

^ + v-Vv = VQ - -Vp. (277) 

3^ p 

86. Nature of the Solution of Partial Differential Equations. In 

7 the meaning of the solution of an nth-order ordinary differential 
equation was explained. To solve such an equation means to find a 
function of the independent variable which satisfies the differential 
equation and which contains n arbitrary constants. The boundary 
(initial) conditions consist of specified values of the function and of the 
first (n 1) derivatives for some definite value of the independent 
variable. 

The solution of a partial differential equation is similar except for 



222 MATHEMATICAL PHYSICS OF VECTOR FIELDS 

the boundary conditions. The number of independent variables is at 
least two. To construct a solution it is then necessary to find a func- 
tion of the independent variables, which when substituted in the 
partial differential equation satisfies it. In general, there is an infini- 
tude of such functions, but only one of these will also satisfy the 
boundary conditions. The boundary conditions in two great classes 
of partial differential equations (including most of those of mathe- 
matical physics) are as follows. The value of the solution must reduce 
to a prescribed function over a surface boundary, or the normal deriv- 
ative of the solution must reduce to a prescribed function over a 
boundary or satisfy certain other conditions. 

In solving a partial differential equation which holds throughout 
a region under consideration the equation in vector form is trans- 
formed into a scalar equation. The coordinate system is chosen such 
that, when one of the independent variables is set equal to a certain 
constant, the boundary of the region is obtained. For example, if 
the region, in which V 2 V = is valid, is a rectangular parallele- 
piped, then Laplace's equation is written 



and the plane x = xo is a portion of the boundary of the region. If 
the region is a cylinder, ellipsoid, sphere, or tore then the coordinate 
system chosen must be respectively cylindrical, ellipsoidal, spherical, 
or toroidal orthogonal coordinates. These coordinates are much 
used in the solution of partial differential equations. 

At the beginning of Vol. II, Chap. Ill, on the solution of partial 
differential equations, a general method is given for transforming 
a partial differential equation into one of the infinitely many systems 
of orthogonal coordinates. 

In 18, a method of solving systems of simultaneous ordinary 
differential equations was explained. By manipulation one differ- 
ential equation was obtained in one of the dependent variables of the 
system. A system of partial differential equations is handled in a 
similar manner. This is illustrated in the derivation, from Maxwell's 
field equations, of the equation of the propagation of electromagnetic 
waves. 

87. The Partial Differential Equations of Electromagnetic Waves. 
The type of general problem described by Eqs. (272) and (273) is as 
follows. A system of moving charges and varying currents exists in 
a varying configuration. It is required to find the H and E fields at 



EQUATIONS OF ELECTROMAGNETIC WAVES 223 

time /. Once H and E are obtained, any question can be answered 
regarding the system. For example, it can be shown easily (Ex. 3) 
that the force on a charge q moving in an electromagnetic field is a 
function of H and E, namely, 



F = q I E + - X B 1, (278) 

where B = /*H. 

As in 18, Eqs. (273) are, in general, not solved directly but replaced 
by two other equations; one in H alone, the other in E alone. This 
is done as follows. Taking the curl of the third equation of (273), 
we have 

V X (V XE) = -^vXH=- ^VXH. (279) 

The partial derivative with respect to the time of the last equation of 
(273) gives 

!;(VXH),^. (2 8 0) 



Substituting the value of (V X H) from (280) in (279), there is 

V X (V X E) = - ~ |^-, 
or, since V-E = 0, 

(See 76, formula 5.) Similarly, 

V 2 H = ^j|^-. (281) 

Eqs. (281) are the equations sought. Applications of these equations 
in the solution of problems of value are found in Ref. 6. 

EXERCISES 

1. Show that the first and third definitions of divergence given in 74 are equiva- 
lent. By Gauss's theorem 

fv-Vdv Cv-dS. 
J J & 

vol 

By the theorem of the mean from the calculus 



ff(x, y> 



vol vol 



224 VECTOR FIELDS 

where (XQ, yo, ZQ) is some point of the volume. Applying the last equation to the 
one preceding it 



V-V(x Q ,y Q ,z Q )fdv = V-V(* , y , o) vol. 



fv-ds 

V.V = lim^- <. 

vol-0 VOl 

2. Show, by means of Stokes's theorem and the method of Ex. 1, that the first 
and third definitions of curl in 75 are equivalent. 

3. Prove that the force on a charge q moving in an electromagnetic field is 

F = q\E + - XBj. 

4. Prove Gauss's, Green's, and Stokes's theorems. 



Ill 

VECTOR FIELDS 
(Vector Magnetic Theory) 

In the design of electrical apparatus, it is frequently necessary to 
know the magnetic flux density not only in the neighborhood of the 
winding but also within the conductors themselves. The flux density 
can be computed, at points exterior to the conductors, from the scalar 
magnetic potential. This is not the case at points within the con- 
ductor because the scalar magnetic potential does not exist inside 
current-carrying regions. The flux density can, however, be computed 
from a mathematical expression called the vector magnetic potential. 
One of the purposes of this section is to explain those portions of vector 
magnetic theory that pertain to the vector potential. Since a number 
of important engineering papers have been written employing the 
vector potential, a second purpose of this section is to furnish the 
background for these papers and to correlate their important concepts. 
Such will furnish the starting point for further investigations in vector 
potential of more and more complicated regions. 

As in Section II of this chapter certain mathematical proofs are 
reserved for exercises. These are found at the end of the section. 

88. Experimental Basis of Magnetic Theory. There are a number 
of consistent logical systems of magnetic theory, and the subject is 
so well developed that many approaches are possible. An approach 
is desired which is primarily an engineering one and also such that the 



FORCE BETWEEN MOVING CHARGES 225 

idea of vector potential and related concepts may be reached as quickly 
as possible. Because electrical engineering is greatly concerned with 
the interaction of electric currents we shall base this section upon 
Ampere's fundamental law. The following relationships hold only for 
steady currents or, what amounts to the same thing, configurations 
of moving charges which may vary from point to point, but which at 
any given location do not, on the average, vary in time. The dis- 
cussion of magnetic vector potential is further restricted here to the 
case of non-magnetic media, i.e., regions characterized by unit perme- 
ability. Ampere's experiments were made with complete circuits, or 
at most dealt with the force between one complete circuit and the 
movable element of another. The following law for the force between 
circuit elements is equivalent to his results, provided it be used to find 
the force between a complete circuit and a circuit element 

//' 
dS = ds X (ds' X r), (282) 

where dF is the force exerted by the element of length ds' t in which 
current /' flows, on the element ds in 
which current / flows. The current is 
positive if positive charge flows in the 
direction of the element. The vector r FIG. 51. 

is drawn from ds' to ds as shown in Fig. 

51. This and succeeding equations are written for quantities ex- 
pressed in c.g.s. electromagnetic units. 

89. Force between Moving Charges. Let us express (282) in terms 
of moving charges, making use of the fact that the current at any 
point in a conductor is the charge passing that point per unit of 
time. If an element ds contains n charges e moving with drift 
speed v, 

_ nev 
d? 
or 

Ids = nev. 

Thus a single charge e moving with a velocity v is equivalent to a cur- 
rent element Ids if 

Ids = ev. 

Thus from (282) the force between moving charges e and e' is 

a^r 

F = X ( V X f )' 



226 VECTOR FIELDS 

where r is the vector distance from e' to e. This may be expressed in 
terms of the field vector B due to e' as 

F = ev X B, (283) 

where 

B . *-'. (2S4) 

The magnetostatic force on e due to a number of moving charges 
e'i, 6*2 . . e'n is still given by (283) if now 

., (285) 



90. Vector Magnetic Potential. In the same way that it is con- 
venient in electrostatic problems to define a potential function from 
which to calculate electric intensities, it is convenient in magnetic 
problems to set up a vector magnetic potential A, from which to calcu- 
late the magnetic flux densities B; the vector magnetic potential 
exists both within and exterior to current-carrying regions. 

The vector potential is defined so that 

B = V X A. (286) 

By the second definition of curl, Eq. (247), the vector identity 

CiVi r t 

V X = dVi X -^ 
Ti r? 

is readily established. From the last equation and the definition of 
B, (285), it follows that 



B = Z, v x = VX 2. . 

Now if we define the magnetic vector potential A by the equation 

A - 2 v < 287 > 

then 

B - V X A. 

Thus when A has been determined for a circuit, the flux density B is 
obtained by a routine process, namely, by the application of the curl 
operator, Eq. (246). 

91. Integral Definition of Vector Potential. Since the vector 
potential is to play such a fundamental r61e in magnetic theory, it is 
important to know its properties. It has been defined in Eq. (287) 



INTEGRAL DEFINITION OF VECTOR POTENTIAL 227 

for groups of charge, but its use, of course, depends upon expressing 
it for ordinary bodies carrying currents. The form of the vector 
potential is perfectly general and may be applied to any sort of moving 
configuration of charge. However, if we here restrict the motions of 
charges to translation only that is, if we consider only elementary 
currents excluding current whirls then (287) may be written in terms 
of current density as 



A = / J , (288) 

\J T 

Vol 

where the integrand is current density in the volume element divided 
by the distance of the volume element from the point at which A is to 
be found. The definition (288) is applicable only to finite bodies. In 
93 this definition is modified to take care of straight conductors of 
infinite length. 

Since B = V X A, it follows from Eq. (4), 76, that V-B = 0. 
This implies that tubes of magnetic flux are always closed on them- 
selves or that flux lines are closed loops. It is true also that, if there 
is no " heaping up " of charge in the body, V-A = 0. Consequently, 
similar closed loops for A can be constructed. 

92. Partial Differential Equation Definition of Vector Potential. 
The vector potential of a finite body may be found by definition (288) 
provided the volume integration can be performed. It is frequently 
easier to obtain the vector potential by the solution of a partial dif- 
ferential equation. 

In Vol. II, Chap. Ill, the following important theorem in the 
theory of partial differential equations is proved. Theorem: A solu- 
tion, continuous with its first derivatives and vanishing at infinity as 
1/r, of the equation V -W = - 4*p (289) 

is given by 



/ , , , 
= / , 

*/ 7 

Vol 



V f = , (290) 

*/ 7 

Vol 

where V is a potential function corresponding to the density p. If p 
is density of electric charge, Vis electric po- 
tential; if p is mass density, V is gravitational 
potential. 

The notation is made clear by Fig. 52, in 
which Q is the moving point of the element pic. 52. 

of integration and P the fixed point at which 
the potential V is to be found. The density at Q is denoted by 




VlSLTl'UK MtLL>5 



If Eqs. (289) and (290) are written in rectangular coordinates, we 
have 



v = f f f e'( x '< y'' ^dx'dy'dz' 

P J J J ((x- x') 2 + (y- y') 2 + (z - z'Y\* ' 

If V and p are replaced in (289) and (290) by A and j, and certain 
boundary conditions taken care of, then the following equivalent 
definition of the vector potential for finite bodies may be written. A 
function satisfying, in regions of unit permeability, the conditions: 



(a) V 2 A = - 47rj, 

(b) A is continuous, 

9A 9A 

(c) - -- h - = 0, 



(d) A is regular at oo , . 



(291) 



is the vector potential defined by (288) in case there is no surface 
magnetization at the boundary of the conductor or in the neighboring 

3A 

medium. The is the derivative of A in any direction n, and n\ 
dn 

and H2 are respectively the interior and exterior normal directions to 
a surface. If there is surface magnetization, condition (c) must be 
modified. Condition (c), as stated above, means that the normal 
derivative is continuous at boundaries between substances both of 
unit permeability, such as air and copper. It is not continuous at an 
iron boundary. (See Ex. 2 96.) 

93. Vector Potential for Infinite Conductors. As will be shown 
below, the vector potential A at point P due to the current in an 
infinitely long straight conductor is given by 



A =-J J jlogrfo, (292) 

where S is the cross-section in the plane perpendicular to the axis of 
the conductor including point P, r is the distance from P to the element 
ds at which the current density is j. This naturally raises the ques- 
tion, how is Eq. (292) related to the definitions of (288) and (291)? 
Definitions (288) and (291) are most used in engineering. (See Refs. 
35-36 at end of text.) 



VECTOR POTENTIAL FOR INFINITE CONDUCTORS 



229 



(0,0, <J t ) 



Az' 



It is the purpose of this article to show the derivation of (292) 
from (288). Suppose that an in- 
finitely long wire coincides with 
the 2-axis as in Fig. 53. By a wire 
is meant a conductor of infinites- 
imal cross-section, say ds. Let the 
current density in the wire be j 
(which is in the z direction), then 
the current in the wire is i = jds. 
Consider a segment (finite length) 
of the wire. Let the ends of 
the segment be C\ and 2- Find 
the vector potential at the point 
P(x, z) by (288). The potential at P is 



P(x,o,z 



FIG. 53.- 




-Vector Potential of Infinite 
Wire. 



/\dz' 
T-' 



or 



A = i log - 



(C 2 - 



- z 



The equipotential surfaces about the segment can be shown to be 
ellipsoids with C\ and 2 as foci. In connection with the study of 
fields about conductors, we desire for simplicity equipotential surfaces 
which are cylinders. Consequently, we let Ci * oo and 2 > o , 
and the ellipsoids approach cylinders, provided the point P(#, z) and 
the direction of the line remain fixed. As C\ and 2 become infinite 
in the manner indicated, the expression for A becomes 



A^ = lim i log 



oo. 



This is an unsatisfactory result. Hence we shall not take Eq. (288) 
to be the definition of potential when the body extends to infinity. 
We observe that, if the body is enclosed in a finite volume, the zero of 
the potential, as given by Eq. (288), is at infinity. We now lead up 
to a definition of potential in the case in which we are interested. By 
rationalizing the denominator in the last fraction, we have for the 
potential of the finite segment, 



(C 2 - 



(C 2 - z)] 



+ log [ V* 2 + (Ci - z) 2 - (Ci - 



230 VECTOR FIELDS 

This value of A is the potential according to Eq. (288) at the point 
P(x, z) of a finite wire segment. Subtract from this value A a con- 
stant K (i.e., a constant so far as the point P(x, z) is concerned, but 
not constant with reference to C\ and C^). For example, let K be the 
potential at the point (a, b) due to the finite wire. The potential at 
P(x, z) due to the infinite wire then is defined to be the previous 
value minus K, or now 

A = i {log~ + log [Vx 2 + (C 2 - z) 2 + (C 2 - z)] 



+ log [Vx 2 + (Ci - z) 2 - (Ci - z)} 
- log-^r - log [Va 2 + (C 2 - b) 2 + (C 2 - b)] 



- log [Va 2 + (Ci - b) 2 - (Ci -b)]}. 
Taking the limit as Ci > oo , C 2 , we obtain 



-. 

When x = a, i.e., on a cylinder of radius a about the wire, A = 0. The 
zero of A is then somewhat arbitrary, depending upon the value of a. 
If for simplicity a is taken to be unity 

A = i log = - i log x 2 . 

This definition, which is the usual one, of the potential of an infinite 
wire is a natural one, since it is the potential (slightly modified) of a 
segment of a wire as the segment becomes infinite. The modification 
imposed changes the zero of the vector potential from infinity to the 
distance a. If we do not choose a zero point of the potential A, then 

A == i log + an arbitrary constant. 
x 

To obtain the vector potential of an infinite conductor, we sum, 
by means of the integral, the vector potential of the wires of infini- 
tesimal cross-section. Hence, 




A - - Jlogr 2 dS, 

where r, Fig. 54, is the distance from the element dS to the point P(r). 



ENGINEERING EXAMPLES 



231 



In finding the vector potential of an infinite straight wire carrying 
a current of density j, the vector potential is seen to be independent 
of the z-coordinate of the 
point P. (See Fig. 53.) Con- 
sequently if Eq. (291) is 
written in rectangular coor- 

3 2 ^ 
dmates the term is zero 

as 2 

and V 2 A = 4?rj becomes 





P interior 



P exterior 



FIG. 54. Vector Potential of Infinite 
Conductor. 



(293) 



94. Engineering Examples. Let it be required to find, both by 
evaluating the integral in Eq. (292) and by solving the partial dif- 
ferential equation (291), the 
vector potential due to an 
infinitely long straight con- 
ductor carrying a current of 
density j. The cross-section 
of the wire is circular, of 
radius a. Let the zero of the 
vector potential be on the 
unit circle, the axis of the wire being the z-axis. 

First method. Let A fl denote the vector potential at an exterior 
point. Then 




FIG. 55. Exterior Vector Potential. 



A e =- 






j log r 2 dS 



j / /log (d 2 + p 2 - 2pd cos 0)pdpdd 

/0 /D 

n 2 T / 2p P 2 \ i 

log d 2 + log 1 cos 6 + ) \pdpdO 
L \ d < 






= jira 2 log d 2 (see Ex. 1), 



or 



where I is the total current flowing in the conductor. If P is interior 
to the cross-section, then 

A '' = ~~ J L/ f l 8 r2dS + 



f f l 



232 VECTOR FIELDS 

The value of the first integral is 2irjd 2 logd. The value of the 
second integral is 

- 2ira 2 ] log a + 2ird 2 j log d + ira 2 j - ird 2 j. 

(See Ex. 1.) Thus 

A, = 7rj(a 2 - d 2 ) - 2<ira 2 j log a. 

Second method. The same result is now obtained by solving the 
differential equation VA 2 = 4?rj or V 2 A^ = accord- 
ing as the point P is interior or exterior to the 
current-carrying region. Since the conductor is 
circular in cross-section, we express the differential 
equation in cylindrical coordinates. By the rules of 
the calculus for change of independent variables, Eq. 
(293) becomes 




FIG. 56. Interior 
Vector Potential. 



1 3 / 
7 f r \ r 



We have found A to have the direction of j (along the z-axis). Con- 
sider only the z component of A. Outside the conductor j is, of 
course, zero and the equation to be solved is 



- I r^-^J = 0. 
r 3r \ Qr / 

Integrating with respect to r, we have 

T = C. 

Integrating again, 

A Ze = Clogr + A. (295) 

For points inside the conductor Eq. (294) must be solved. Since A 
and j have the same direction, Eq. (294) may be written 



1 3 / 3A,\ 

~ I r * I =- 4?r;. 

r dr \ Qr / J 



dr 

The current density j is constant over the cross-section. Integrating, 
there is 



Integrating again 

A,. = - vr 2 j + d log r + ki. (296) 



ADDITIONAL VECTOR RELATIONS 233 

The constants C, k, Ci, and k\ of Eqs. (295-296) are determined 
subject to conditions (b) and (c) of Eqs. (291), and subject to the two 
additional conditions that A vanish at r = 1 and that A remain finite 
at the origin, i.e., on the axis of the wire. A is known to be finite at 
the origin by definition (292). Applying the last condition first, it 
follows that Ci = 0. If a < 1, we must apply the condition that 
A = at r = 1 to Eq. (295). Hence k = 0. Applying the condition 
that A (or A z in this case) is continuous at the boundary r = a, the 
values of A t given by Eqs. (295-296) must be equal for r = a. That is, 

Clog a = -7r/a 2 + *i. (297) 

Condition (c), namely, 



3A 



9A 



= ' 

r=a ~""~ 



or 



dr dr 
gives the equation 



or 

C = - 2wa 2 j. (298) 

From (297) and (298) it follows that 

ki = Trja 2 2irja 2 log a. 

Substituting in Eqs. (295-296) the values of the arbitrary constants 
which have been determined, we have 

A e = - 7ra 2 j log r 2 , 

A = ?ra 2 j 7ir 2 j 2?ra 2 j log a. 

These expressions for A e and A> agree, of course, with those obtained 
by the definition of Eq. (292). 

95. Additional Vector Relations in Vector Magnetic Theory. 
Vector identity (5) of 76 is 

V X (V X F) = V(V F) - V-VF. 
It was noted in 91 that V-A = 0. From 92 we have 
V 2 A = V-VA =- 4rrj. 



234 VECTOR FIELDS 

From these three relations it follows that 

V X (V X A) = 4rrj. (299) 

But V X A = B. Hence 

V X B = 4rrj. (300) 

96 . Summary. To obtain the flux density in current-carrying 
regions (and non-current-carrying regions) compute the vector poten- 
tial by either of the definitions (288), (291), or (292). Take the curl 
of A. This is B the flux density. If the coordinates used are rectan- 
gular coordinates, it is only necessary to apply Eq. (246). If the 
coordinates are cylindrical, spherical, or any other curvilinear coordi- 
nates, it is necessary to employ the expression for curl in curvilinear 
coordinates developed in Vol. II, Chap. III. 

EXERCISES 

1. Show that 



--f c 8 * + T <tf = Ofor [-} < 1 
d a 1 \d 



First from 



prove 



and 



( e te\ / ,-w\ 
i - -Hi -) 

I r 2 r 3 \ 

log(l - 2rcos0 + r 2 ) = - 2( r cos d + - cos 20 + - cos 36 + . . . j 

/cos d cos IB cos 3d \ 

log (1 - 2r cose + r'-) =+ 2 logr - 2^ + + + . . .J. 



2. For a discussion of the behavior of B and A at boundary regions between sub- 
stances of different permeability read: 

(a) " Fundamental Theory of Flux Plotting," Stevenson, General Electric 
Review, Vol. XXIX, November, 1926. 

(b) The Electromagnetic Field, Mason and Weaver, p. 208. 

3. Obtain the vector potential for an infinite conductor of square cross-section 
and constant current density j. 

4. Show that - + ~ = 4*7 in rectangular coordinates reduces, by change 
of dependent variables, to 

I r 1 " 4*-;' in cylindrical coordinates, if A is not a function of 0. 

rdr\ dr/ 



DIGRESSION FROM DEFINITIONS 



235 



IV 
DYADICS 

This introduction to dyadics is directly preparatory for the study 
of the General Electric Company text, An Analysis of Synchronous 
Machines, and for the derivation of the equations of elasticity. This 
brief treatment is divided into three parts: 

(a) Certain formal definitions. 

(b) A number of necessary theorems. 

(c) Elementary applications. 

97. Definitions. The symbol ab (i.e., two vectors placed in 
juxtaposition) is called a dyad. An algebraic sum of such terms 
(ab + cd + ef + . . ,) is called a dyadic. Since any dyadic can be 
reduced to the algebraic sum of three (or less) dyads, it is necessary 
to discuss dyadics only of the form ab + cd + ef. A dyadic is a 
mathematical operator having no physical significance in itself, but the 
important operations performed by this operator have physical sig- 
nificance. 

98. Digression from Definitions to a Physical Example. To see 
in one case, at least, how dyadics arise, consider the following problem. 
Let a deformable body be subject to a homogeneous strain. Let it 
be required to express the displacement of a general point P as a 
function of the vector posi- 

tion of the point prior to 

the strain. A body is de- 

formable if its particles are 

capable of displacement rela- 

tive to each other. Such a 

relative displacement is called 

a strain. Let P(x, y, z) de- 

note the location of a par- 

ticle of the body prior to the strain, and P'(x', y', z') the location 

of the same particle after the strain takes place. In case of a 

homogeneous strain, the coordinates x 1 ', y 1 ', z' of P' are linearly expres- 

sible with finite scalar constant coefficients in terms of the coordinates 

jc, y, z of P. By reference to Fig. 57, this definition is expressed by 

means of the equations 

x' = anx + any + 




FIG. 57. 



y' = 



(301) 



236 DYADICS 

Evidently from the figure 

r = ix + jy + kz, 

r ' = ix' + j y ' + kz'. (302) 

In view of Eqs. (301), Eq. (302) is 
r' = i(anx + any + #132) + j(a 2 ix + a 22 y + a 23 z) 

+ k(a 3l x + a 32 y + 033*) (303) 

= iai-r + ja 2 -r + ka 3 -r, (304) 

where 

+ jai2 + kai3, 



a 2 = 



(305) 



Eq. (304) may be written in the form 

r' = (iai + ja 2 +ka 3 )-r, (306) 

where the right side of (306) is only a symbol denoting the same as 
the right side of (304). We have thus obtained in (306) a new symbol, 
the dyadic. If from each side of (304) the vector r is subtracted, then 



s = r' r = iai -i + ja 2 -r + kas-r ii-r jj-r kk-r 
= (iai + ja 2 + ka 3 ii - jj - kk) -r. 



(307) 



The algebraic sum of the six dyads on the right side of the equation 
is a dyadic. Thus (307) may be written 

s = i^r, (308) 

where ^ is a dyadic. Thus the symbol ^-operating on the position 
vector r yields a displacement vector s. We now return to the subject 
of definitions and lay down such further ones that from them we may 
establish theorems which in turn are useful in the application of 
dyadics to physical problems. 

99. Definitions Resumed. The dyadics ab + cd + ef and ba + 
dc + fe are conjugates. 

The dot product of a dyadic ^ = ab + cd + ef into a vector r is 
defined by the equation 

^r = ab-r + cd-r + ef -r = S ab-r. 

Likewise r-^ = S r-ab. From these two definitions, it follows that 
in general ^-r j& r-^. 



THEOREMS 237 

The cross product of a dyadic \t/ into a vector r is defined by the 
equation \j/ X r = 2 ab X r. Two dyadics ^ and </> are said to be 
equal if and only if 

r-0 = r-^ (309) 

and 

-r = *-r (310) 

for all values of r. In the dot product r-^ the dyadic is called a post 
factor; in \l/-i the dyadic \l/ is called a pref actor. If a dyadic <t> is such 
that <-r = r and r-< = r for all values of r, then </> is an idemfactor, 
usually denoted by /. If all the vectors in the dyadic < = ab + cd 
+ ef are expressed in terms of their i, j, k components and the results 
expanded there results 



iji + #22 jj + 023 jk 
a 3 iki+ # 



where the a's are scalar constants. This form of </> is the nonion form. 
The nonion form of the idemfactor is / = ii + jj + kk. 

100. Theorems. The theorems desired are written as the following 
equations: 

I. t-(Ti + r 2 ) = ^-ri + \p-r 2t 

II. r-(^i + ^2) = r-ti + r-ife, 

III. (a + b)c = ac + be, 

IV. 0-(r X s) = (0 X r)-s, 

V. a Xr = (/-a) Xr = (/Xa)-r =~r-(/Xa). 

To establish theorem I, apply the definition of the dot product and 
obtain 

t- (ri + r 2 ) = Sab - (ri + r 2 ) 

= 2ab-ri + 



which completes the proof. We leave the proof of II, III, and IV as 
exercises but establish V. By the definition of idemfactor /, it is 
true that /-a = a. Obviously a X r == (/-a) X r. To see that 



238 



DYADICS 



(/a) X r = (/ X a)-r it is necessary only to reduce both expressions 
to a X r. Now 



(7-a)Xr = aXr 
(/ X a)-r = S (ii X a)-r = Si(i X a)-r 



and 



= S i(i-a) X r = /-a X r = a X r. 
This concludes the proof. 

101. Applications. Both the theory and applications of dyadics 
are as extensive as the theory and applications of vectors. Many 
analogous theorems hold in the two subjects. We are interested in 
only three elementary applications. The first, with reference to 
elasticity, has already been given. The second is concerned with the 
rotational property of dyadics. 

(a) Rotational dyadics. In Fig. 58, if a is a unit vector then the 
operator a X applied to the vector r perpendicular to a turns r through 
a right angle. This follows from the definition of a cross-product. 
But by theorem V, a X r = (/ X a)-r. Hence (/ X a)- is an 
operator which rotates the vector r through a right angle about the 





FIG. 58. 



FIG. 59. Direct and Quadrature Axes. 



line a. If again (/ X a) be applied to the vector (/ X a) -r the vec- 
tor (/ X a) -i is turned through 90 degrees or r is then turned through 
180 degrees. If r is parallel to a, then (/ X a) annihilates r since in 
this case r X a = 0. If r is neither parallel nor perpendicular to a, 
the (/ X a)- annihilates that component of r parallel to a and rotates 
through 90 that component of r perpendicular to a. Let the unit 
vectors d, q, and z be directed along the x, y, and z axes respectively. 
(This notation is useful in synchronous machine theory where q is in 
the direction of the quadrature and d in the direction of the direct axis.) 
The idemf actor then is / = dd + qq + zz. Let r be any vector per- 
pendicular to the 0-axis. 

Then (/ X z)-r = [(dd + qq + zz) X z]-r = (qd - dq)-r. But 
by theorem V, we have 

(JXz)-r = z Xr, 



APPLICATIONS 239 

where z X turns r through 90 degrees. Since the operator V 1 
performs the same operation on complex numbers, it follows that 

v 1 is equivalent to (qd dq), 

each in its own system of representation. This is a relation of frequent 
occurrence in some treatments of synchronous-machine theory. 

(b) Impedance as a dyadic. Let the unit vectors i, j, k be taken 
along the x, y, z axes. Electrical impedance, r + x\/ 1 in complex 
number notation, may then be written 

z = ir + jx 

= r(ii + jj)-i + (ji - ij)-i* 
= r/-i + *(k X 7)-i 



where / is the idemf actor ii + jj. 



EXERCISES 

1. If g is conductance and b susceptance, express admittance as a dyadic operator. 

2. Given that 

- Xd(p}id, 



and 

ed == ejd, e q = <&, id = idd 

e = ed + e 4l i = id + ij, etc., 

where d and q are unit vectors in line with the direct and quadrature axes, respectively 
Show that 



where 

g(P) = &(P)dd -h 



CHAPTER IV 
HEAVISIDE'S OPERATIONAL CALCULUS 

The Heaviside operational calculus is, to both electrical and 
mechanical engineers, one of the most valuable branches of mathe- 
matics. There are, at present, three main approaches to this subject. 
The first is by means of line integrals in the complex plane, the second 
by Laplace's integral equation, and the third by operator-experimental 
processes. A general approach is desired so that the student has not 
only a tool by which a few particular well-known exercises can be 
solved, but also an instrument of research by which he can make difficult 
investigations. The line integral method satisfies this requirement. 
This approach has the disadvantage that the student is delayed 
in the study of the operational calculus until certain theorems on 
line integrals are understood, but the subsequent rapid progress, 
due to more powerful methods, compensates for this delay. The 
only knowledge presupposed for understanding this chapter is the 
definition of a definite integral from the calculus, the elementary prin- 
ciples of circuits explained in 19-20, and the theory of determinants 
found in 21-27. 



PRELIMINARY MATHEMATICAL PRINCIPLES LINE INTEGRALS IN THE 

COMPLEX PLANE 

The line integral theorems required are : Cauchy's first and second 
integral theorems, Laurent's theorem or expansion, and the residue 
theorem. But before understanding these theorems, in preparation 
for the study of the operational calculus, let us first answer the natural 
questions: what kind of engineering problems does Heaviside's oper- 
ational calculus solve, and what is the history of the development of 
this subject concerning which there has been so much controversy. 

102. Some Engineering Problems Solvable by Operational Calcu- 
lus. The operational calculus is of greatest use in obtaining transient 
responses of electrical and thermal circuits and of mechanical systems 
to suddenly impressed voltages, heat densities, and forces, respectively. 
It will, however, give also the steady-state response. The value of 

240 



SOME ENGINEERING PROBLEMS 241 

this calculus is quickly displayed by the general statement of engineer- 
ing problems which it readily solves. 

(a) Electric circuits with concentrated (lumped) parameters. Let 
there be given a linear network of n meshes (branches) with concen- 
trated circuit parameters. Let the n meshes be coupled in any or all 
of the ways described in 20. Suppose the n meshes are in a state 
of equilibrium, that is, no currents flowing or charges existing. It is 
required to find the response (currents in each branch) of the network 
when a constant voltage E is suddenly applied in any mesh. 

Suppose that, instead of a constant voltage, a voltage which is a 
function of the time is suddenly applied. Find the response. 

Finally let n variable voltages be applied, one in each branch of 
the network. Again, find the response. 

If, instead of applied voltages, we have suddenly applied currents 
in the meshes of a network the operational calculus will yield the 
voltages induced across any element of the network at time t sub- 
sequent to the application of the currents. 

In certain cases the response can be calculated even if the network 
is not in equilibrium when the voltages are impressed. 

(b) Transmission line. Consider a transmission line possessing 
distributed inductance, resistance, capacitance, and leakage. Sup- 
pose in addition there is a concentrated impedance at both the sending 
and receiving ends of the line. Let it be required to find the current 
at any time / after a voltage has been suddenly impressed at the send- 
ing endof the line. 

(c) Linear heat flow. Consider a flat wall consisting of alternate 
layers of metal and insulating material. Let the wall be of sufficient 
area that a flow of heat within the wall may be considered to flow in a 
direction perpendicular to the wall. 

Suppose that a definite amount 
of heat is given to each unit area 
per unit time. Find the tempera- 
ture at any point of the wall at any 
time /. 

(d) Condenser-type Thyratron 

inverter. A constant voltage E is FIG. 60. Thyratron Inverter, 

applied to the Thyratron inverter 

circuit as shown in Fig. 60. The circuit is arranged so that one and 
only one tube is firing at any time. Obtain transient and steady- 
state load current. (See Ref. 6.) 

(e) Seismographs. The operational calculus is a natural tool 
for the investigation and interpretation of the motions of seismographs. 




242 PRELIMINARY MATHEMATICAL PRINCIPLES 

In this case the suddenly impressed quantities are either velocities 
or accelerations. The reader is referred to Vol. II. 

103. Historical Note. Oliver Heaviside lived during the perio 1 
1850-1925. One of his many contributions to knowledge of electrical 
phenomena was the operational calculus with which his name is always 
associated. His efforts in this field were but slowly appreciated, 
partly because his results were not obtained in a mathematically 
rigorous way. The formulas derived by him yielded correct results 
in almost every case, but his justification of them was not pleasing to 
certain of his mathematical and engineering contemporaries. How- 
ever, in 1916, T. J. FA. Bromwich in England and K. W. Wagner in 
Germany placed the Heaviside operational calculus on a mathemati- 
cally rigorous foundation by means of the theory of functions of a 
complex variable. In 1925, John Carson also gave rigorous proofs, 
by means of integral equations, of Heaviside's methods. In 1927, 
H. W. March showed that Bromwich 's integral is a solution of Carson's 
integral equation. These papers form the framework of the rigorous 
theory of the operational calculus. References to these and other 
important papers on the subject are found at the end of the text. 
// should, of course, be remembered that Heaviside was the discoverer or 
inventor of the subject which bears his name, and that the subsequent 
rigorous proofs, although of great value, are in a rather definite sense only 
an improvement. 

If the student has as much knowledge of the theory of functions 
as is given in Vol. II, Chap. IV, it will be time saved to omit* 104- 
114 and proceed at once with the operational calculus in 115. On 
the other hand, no knowledge of function theory is 4 presupposed in this 
chapter, and the account of line integrals in 104-114 is ample for 
the understanding of Bromwich 's rigorous results. The remainder of 
this section is devoted to the explanation of the requisite theorems on 
line integrals in the complex plane. 

104. Complex Numbers and Functions. Although a knowledge 
of the properties of and elementary operations with complex numbers 
such as representation by Argand's diagram, addition, multiplication, 
division, extraction of roots, and De Moivre's theorem for n a positive 
integer is presupposed, it is wise to review these subjects briefly. 
Just as all real numbers may be represented by points along the #-axis, 
so complex numbers (i.e., numbers of the form x + iy where x and y 
are real numbers and i =\/ 1) may be represented by points in a 
plane. The old x-axis and y-axis of analytic geometry become respec- 
tively the axis of reals and the axis of imaginaries. The point x + iy 
is then plotted as shown in Argand's diagram, Fig. 61. The real num- 



COMPLEX NUMBERS AND FUNCTIONS 243 

bers p = + -\/x 2 + y 2 and = arctan y/x are called respectively the 
modulus and the amplitude of the complex number z = x + iy* Evi- 
dently, from the figure, 

z = p(cos + i sin 0). 

In elementary algebra it is shown that if z\ = pi(cos 0i + i sin 0) and 
22 = P2(cos 2 + i sin 2 ), then 

2i2 2 = pip 2 [cos (0i + 2 ) + i sin (0i + 2 )], 
- = - [cos (0i - 2 ) + i sin (0i - 2 )]. 

2 2 p 2 

It is also shown that 

z n = [p(cos + i sin 0)] n = p n (cos nO + i sin nO), 

where n is any positive integer. The last relation is De Moivre's 
theorem. 

The complex quantity z = x + iy, where x and y are independent 
real variables and i = \/ 1> is called a complex variable. Any 
expression of the form U(x, y) + i V(x, 
y) is a function of z = * + iy, if, when ***'"*'"* 
# and y are given, at least one value of 
U and V are known. If no restrictions 
are placed upon U and V, the study 
of functions of the form U(x, y) + 
iV(x, y) becomes nothing more than FIG. 61. 

the study of pairs of real functions, 

the i fulfilling no purpose. If, however, proper restrictions are placed 
upon U and V, there results a subclass of functions which possess 
many of the properties of real functions and permit the development 
of a calculus called the theory of functions of a complex variable. 
These restrictions appear later. 

In the study of continuous, single-valued, real functions, y = y(x), 
defined in the interval x\ g x ^ # 2 , the graph of y = y(x) is a con- 
tinuous curve. As the independent variable x ranges over the interval 
xi ^ x ^ X2 t the point P(x, y) describes the curve and the ordinate y 
of the point P(x t y) gives the value of the function. 

In the study of continuous single-valued complex functions, such 
as W = f(z), for each value of the independent variable z = x + iy 
corresponding to a point in the z-plane, there is a value of W = U + iV 
corresponding to a point in the W-plane. For example, suppose 




244 



PRELIMINARY MATHEMATICAL (PRINCIPALS 



W = z 2 . Let z have the value 1 + \/3*, then W = (1 + 
= 2 + 2\/3i, and we have the representation as follows: 





FIG. 62. 

And in general if z takes on values within some region such as R z ,for each 
value of z there will be a corresponding value of W lying in the region 
R w in the W-plane. 

A function W = f(z) is said to be continuous at the point z if to 
every positive number 8 another positive number 5 can be found, 
such that 

whenever | h \ < d. This is the same as the definition of continuity 
of a real function except that the absolute value of h replaces the 
positive quantity h. Obviously if /(z) = U(x, y) + iV(x, y) is a con- 
tinuous function, then both U and V are continuous functions. 
A regular arc is defined to be a set of points such that 

1. x = /(/), y = g(f) where / is a parameter and t\ g / g fa. 

2. /(Oi #Wt /'W an d g'(t) are single-valued and continuous for 
all values of / for which t\ ^ t ^ fa. 

3. [/'(OP + k'(/)] 2 ^ o. 

A regular curve consists of a finite number of regular arcs which are 
joined end to end. A regular curve is said to be closed if /(/i) =f(fa) 
and g(/i) = g(fa). A portion of the plane R is said to be connected if 
every pair of points in R may be joined by a regular curve possessing only 
points of R. A region is defined to be a connected portion of a plane. 
A closed region, that is, a connected portion of the plane including its 
boundary, is called a connex. A region is simply connected if any 
closed curve in it encloses only points of the region. 

105. Definition of a Line or Curvilinear Integral in the Complex 
Plane. Before beginning the study of curvilinear integrals in the 
complex plane, we review certain principles of the calculus. Let 



DEFINITION OF A LINE OR CURVILINEAR INTEGRAL 245 



y f ( x ) be a real function. Divide the interval a ^ x ^ b into n 
parts of lengths A#i, A#2, A#a . . . A# n . Let 5 be as great as any A#< 
for i = 1, 2, 3 ... n. The positive number 5 is called the norm of 
A#i, AX2, . . . A# n . Let t be a point in A#,-, either an interior or an 
end point. If 



Lim 



)A* n ] = f 
J * 



f(x)dx 



exists, this limit is the definite integral of f(x) from a to b. This limit 
exists for a very large class of functions. 

The definite integral in the complex plane is defined in a similar 
manner. Let the curve MNP, Fig. 63, be divided into n parts by 
the points zi, 22, ... z n . The 
lengths (A*)i, (As) 2> . . . (Az) n , 
where (Az) = z- Z;_i, are di- 
rected chords of the curve. Let 
; be the initial point of the arc 
whose chord is (Az)<. t may be 
chosen as any point on the arc; FIG. 63. 

in the limiting process variations in 

sum values caused by differently chosen **s disappear. Let 8 be 
the norm of (Az)i (i = 1, 2, ... n). If 




Lim 

5-+0 

n oo 



exists, this limit is called the definite integral of /(z) along MNP 
between the limits ZQ and z n . The symbol of this limit is 



f n f(z)dz or f 

* MNP 



f(z)dz. 



In the theory of functions of a complex variable only those func- 
tions of z are considered which can be expressed in the form U(x, y) + 
iV(x, y). We shall show that line integrals in the complex plane can 
be made to depend upon line integrals whose integrands and differen- 
tials are real by means of the formula 



//(*)& = f(Udx - Vdy) + i f(Vdx + Udy), 

Jc */<7 /C7 



(311) 



246 PRELIMINARY MATHEMATICAL PRINCIPLES 

where C denotes the path of integration. Formula (311) is established 
directly from the definition of / f(z)dz as follows. Since 



and 
the sum 

n 

, , + *(Ay),] 



n 



1 

n 



1 
Taking the limit of S n we obtain 

Lim 

n-oo 
S-+Q 

= I (Udx - Vdy) + i I (Vdx + Vdy). 
Jc Jc 

Thus / f(z)dz is expressed as the sum of two integrals whose 

Jc 
integrands are real. The variable in the last two integrals may be 

changed by the equations x = x(t) and y = y(t) so that / f(z)dz is 

Jc 
expressed as the integral of a single real variable t. Or we may make 



in / 
Jc 



the transformation directly in f(z)dz without using Eq. (311). 

EXAMPLE 1. Evaluate the integral / zdz, where C is the arc of 

Jc 

the ellipse joining the points (0, b) and (a, 0). Since U = x, V = y, 
x = a cos /, and y = b sin /, Eq. (311) yields 

/zdz = - / (a 2 + b 2 ) sin t cos tdt + iab I (cos 2 / - sin 2 t)dt 
J*/2 J*/2 



GREEN'S FORMULA 247 

EXAMPLE 2. Evaluate the integral / , where C is the circle of 

JG % 

radius r and center at z = 0. Then z (on path C) = re i$ , dz = ire ie dd 
and 



EXAMPLE 3. Evaluate the integral / (2 zo) n dz, where C is the 

*/C7 

circle whose center is 20 and whose radius is r and n +1, 2, 
3, .... Let z ZQ re {e . Then dz = rie w dd and 

//* 2T - (n+l) (n+l)0 ^ n+1 tn!01 2ir 

*/o ~ i(n + 1) 6 Jo "" 

(To see that e i(n * l}2 * e = 0, it is only necessary to recall from 
the calculus that e ine = cos nO + i sin #0.) 

Let ZQ and 2 n be two points on C, the path of integration. Then 
the following properties of a line integral are easily provable from 
either the definition of an integral or from Eq. (311). 

(a) f f(z}dz =- f f(z)dz, 

2n *^ZQ 

r(**n /*n 

[/i (2) /2(z)]dz = / fi(z)dz / f2(z)dz. 
ZQ *^ 2ft 

106. Green's Formula. The following theorem is needed in the 
proof of Cauchy's theorem. Let R be a simply connected region in 
the plane of reals bounded by a contour C. Let P and Q be any func- 

r T) "r^O 

tions of x and y which together with and are single-valued and 
continuous in ^? and on C. We shall prove : 



This identity, Green's formula, is easily proved by the calculus. Let 
R be divided into subregions /? bounded by C such that any parallel 
to the y-axis cuts the contour in at most two points. Then in RI (say) 

f f^-dxdy = f dx f ' ^dy= f [P( X ,y 2 )-P( X ,yi)}d x . (313) 
JR^ dy J* Jnto dy J* 



248 PRELIMINARY MATHEMATICAL PRINCIPLES 

The last integral is a line integral since y<i and y\ are functions of x. 
Since / P(x, yz)dx = / P(x, y>2)dx, Eq. (313) may be written 

J a */& 

/ / dxdy=-/Pdx or / /- dxdy = f Pdx 

J Rl J dy J Cl J Rl J dy J Cl 




FIG. 64. 



where now the line integral is taken over the closed contour C\ in the 
counterclockwise direction. Similarly, 



f f~dyd X ^ fQdy. 

SR\ J OX */d 



Formula (312) is obtained by adding the last two equations for all of 

the regions and contours Ri and d of R and C. On the common 

inner boundary of two regions Ri (say RI and RI) the line integrals 
taken in opposite directions cancel. 

107. Analytic Function of a Complex Variable. If U and V in 

Eq. (311) satisfy the conditions placed on P and Q in 106, then 
Eq. (311) may be written 



Suppose U and V are subject to the additional restriction that in R 
and on C 



(314) 



9*' 



Then it is evident that I f(z)dz = 0. Eqs. (314) are known as the 

Jc 

Cauchy-Riemann differential equations. It may seem that f(z) = V 
+ iV is greatly restricted. However, infinitely many functions f(z) 



CAUCHY'S FIRST INTEGRAL THEOREM 249 

satisfy all these conditions, and it is in these functions that we are 
interested. This leads us to the definition of an analytic function. 
A function of a complex variable, /(z), is said to be analytic in a region 
Rif: 

(a) f(z) has a definite value for every z in R, 

(b) If the expression \- i exists, has a unique value, and 

QX OX 

is continuous in R. 
It will be shown in Vol. II, Chap. IV that a necessary and sufficient 

condition that h i exist and have a unique value is that U x , 

dx dx 

U v , V x , and V v be continuous in R and Eqs. (314) hold for every 
point z of R. 

EXAMPLE 1. Isf(z) = x + iy, an analytic function? In this case 
U = x, V = y and Eqs. (314) are satisfied. Since f(z) has a definite 
value for every value of x and y and all continuity conditions are 
satisfied, the function f(z) x + iy is analytic for all finite values of 
x and y. 

The function f(z) = x iy is not analytic since Eqs. (314) are 
not satisfied by this function. 

108. Cauchy's First Integral Theorem and Its Corollaries. 
Cauchy's first theorem is an obvious consequence of the definition of 
an analytic function and Green's formula. His theorem is: Let/ (2) 
be analytic over a connex R. Let C be any simply closed curve, the 
boundary of R, or lying entirely within R. Then 



I 



f(z)dz = 0. 



The first corollary of Cauchy's first theorem is: Let C and C\ be 
two simply closed curves such that C completely encloses C\. Let 
/(z) be analytic in the region between C and Ci and on C and C\. 
Then 



/*/()& = f 
Jc Jc 



f(z)dz, 



both integrations being either in the clockwise or counterclockwise 
direction. The positive direction of integration around a closed con- 
tour is defined to be such a direction that the area within the closed 
curve lies to the left of an observer on the curve and facing in the 
positive direction. By 105, reversing the direction of integration 



250 PRELIMINARY MATHEMATICAL PRINCIPLES 

changes the sign of the result. Unless the direction is specified, 
positive direction is assumed. 

The proof of the corollary is as follows: From Cauchy's first 
theorem and Fig. 65, 

( f(z)dz = y/(*)<fc +ff(z)dz +ff(z)dz +ff(z)dz = 0. 

ABDEFQA ABD DE EFG GA 




v_ 

But 
Hence 


-^-"Tr ~- 
FIG. 65. 

jf(z)dz + y 

DE GA 




or, noting directions of integration 

ff(z)dz = //(*)&, 
Jc */Ci 

which is the corollary. 

A second corollary is as follows. If /(z) is analytic in a connex R, 
and zo and z are two points of R, then the value of the line integral 



/ 

'ai 



f(z)dz does not depend upon the path of integration joining 



and z. To see that this is true, apply Cauchy's theorem to Fig. 66, 
where ztfy'z and 20^0 are any two different paths between ZQ and z. 
Since 



=ff(z)dz +f 



f(z)dz +f(z)dz = 0, 



Thus the last integrals are independent of the path and depend only 
upon the end points whose coordinates are ZQ and z. 

109. Definitions of Certain Elementary Functions of a Complex 
Variable. We seek to define some of the well-known functions of the 



DEFINITIONS OF CERTAIN ELEMENTARY FUNCTIONS 251 

calculus for the case of a complex independent variable. We desire 
that the new definitions : 

(a) Reduce to the definitions of the calculus of reals when the 
independent variable assumes real values. 

(6) Give properties to the functions of a complex variable that as 
far as possible are the same as the properties of the same functions of 
a real variable. Conditions (a) and (b) are satisfied if ^, sin z, and 
cos z are defined by the equations 

f = ^'"EE <* (cosy + i sin ?) s 1 + Z + ~ + ~ + . . . , 

6"-<f te Z 3 Z 5 

smZ = ^ =Z--+--..., 

e iz _J_ e -iz Z 2 g4 

co..- -1--+--.... 



The function R = log x is defined in the calculus of reals as the 
inverse of e R = x. If R and x are replaced respectively by the com- 
plex variables W and z, and log z is defined as the inverse of z = e w , 
then W is infinitely multiple-valued. That is, to each value of z, 
except 2 = 0, there correspond many values of W. To see this, write 
z and W in the forms 

z r (cos + isin 0), 

W = U + iV. 
Then 

z = r (cos 6 + i sin 0) = e u + iv = a 17 (cos V + i sin F). 

Equating reals and imaginaries we obtain 

r cos 6 = e u cos F, 

r sin 6 e u sin F. 

If these equations are squared and added, we have 
r = e u or U = log r. 

From the same two equations V = 6. If, then, we define W == log 
z as the inverse of 2 = e w 9 we have 

log z W = U + iV = log r + iQ = log | z | + i amp z. 
But 2 may be designated in the 2-plane by any of the following forms: 

(r, 6 + 2nir), where n = 0, 1, 2, 3, .... 
Thus, log z is infinitely many- valued. 



252 PRELIMINARY MATHEMATICAL PRINCIPLES 

The function z a (a and z real or complex) is defined by the equation 

a __ a log z __ a log \z\ t aid t Zirnai 

110. Integrals and Derivatives of Elementary Functions. In 108 

it was shown that the integral / f(z)dz, where f(z) is analytic, is 

/so 

independent of the path. By making use of this fact and Eq. (311), 

r* 

the value of / f(z)dz, where /(z) is an elementary function, can be 

/BO 

found. For example, let/(z) = e* and let the path connecting ZQ and 
z be the straight-line segments joining ZQ XQ + iyo to x + iyo and 
x -\- iyo to x + iy. The function e z is analytic at all finite points of 
the complex plane. We then have, on the path assigned, by (311) 
and the definition of e 1 , 

~*+ty ~* . r . 

I e g dz = I e x (cos yodx sin yodyo) +i I e x (sin yodx cos yodyo) 

/XQ+iVQ /XQ */XQ 

r v r v 

+ / e x (cos ydx sin ydy) + i I e* (sin ydx cosydy). 

In the first integral on the right y = yo on the path of integration 
from #o + iyo to x + iyo. Hence dy = 0. We thus have 

/e x (cos yodx sin yodyo) I e? cos yodx = cos^o^* e**). 
-v *^ Q 

If the other three integrals are evaluated by the same principles, the 
value of / e'dz turns out to be 

/so 

/e*dz = e*(cos y + i sin y) e XQ (cos yo + i sin ;y ) = e* e*. 
-v 

In the same way, it can be shown that : 

sin zdz = (cos z cos 20), 

i 

cos zdz = sin z sin ZQ, 



f. 
i 



n -j- 1 n 



Z W 

( = + 1 - 2 - 



(315) 



TAYLOR'S SERIES AND SINGULAR POINTS 253 



In the calculus of reals the derivative of f(x)d& with respect to x 

JXQ 

f(x). Similarly, in the theory of functions the derivative is defined 
as the inverse process to that of integration. We thus have from 
Eqs. (315) the relations 

d cos z 

sin z, 



dz 

d sin z 
dz 

d 



= cos z, 



d z n + 1 



= z n . 



111. Taylor's Series and Singular Points. It is recalled from the 
calculus that if /(#) and its first n derivatives exist and are continuous 
over the interval a ^ x ^ b, then f(x) may be expanded in that 
interval in a polynomial of the form 



/(*) =/(a) +/'(a)(*-) +/" (a) + . . . + /""to) , (316) 

L\ n\ 

where/',/", . . ./ (n) are respectively the first, second, and nth deriv- 
atives of f(x) and a < x\ < b. Eq. (316) is Taylor's formula. As n 
approaches infinity in (316), Taylor's formula becomes Taylor's 
infinite series. Taylor's infinite series represents /(tf) for those values 



> 
of x, and those only, for which - - - approaches zero as n 

n\ 

approaches infinity. Taylor's expansion is unique if it exists. It is 
also remembered from the calculus that f(x) = log x cannot be 
expanded in a power series in (x a) where a 0. The point x = 
is a singularity of log x. 

It is proved in Vol. II, Chap. IV, that Taylor's series holds for 
analytic functions of a complex variable. It is 

/(*) = /(a) + /'(*)(* - a) +^y L) (z - a) 2 + . . . . 

When a function is said to be analytic this does not mean that it is 
analytic for every point of the z-plane or for values of z which are 
infinite. In fact, there is a theorem w r hich states that the only func- 
tions analytic for all values of z are constants. A function is called 
analytic if it is analytic in some region. Thus for all functions (con- 
stants excepted), there exist values of 2, finite or infinite, for which 



254 PRELIMINARY MATHEMATICAL PRINCIPLES 

f(z) is not analytic. For these values of z, the function does not 
satisfy the conditions stated in 107. Moreover, it will be shown 
later that there exists no Taylor's expansion at such points, i.e., if 
f(z) is not analytic at a then Taylor's series is not valid. If f(z) can 
be expanded in a Taylor's series which converges for all points about 
a and interior to a circle whose radius is greater than zero then f(z) is 
called a regular function at a, and a is called a regular point of f(z). 
Points in the s-plane which are not regular points of f(z) are singular 
points of /(a). 

112. Singular Points. Poles, points of discontinuity, and essential 
singularities are singular points of single-valued complex functions; 
these same singularities and in addition branch points are singular 
points of multiple-valued functions. If, in the expansion of f(z) in a 
Taylor's series, 

/(a) =/'(a) = . . ./fr-V) =0 
and 

/ (n) * 0, 
then 



..., 

and the point a (real or complex) is said to be a zero of order n off(z). 
The function /(z) in this case may be written 



where g(a) 7* 0. If a is a zero of order n of -, then a is a pole of 

/ W 

order n of /(z), and conversely. Or the definition of a pole may be 
stated in the following way. If f(z) may be written in the form 



where g(z) is analytic at z = a and g(d) 9* 0, then a is a pole of f(z) of 
order n. Essential singularities will be examined in 113 and branch 
points, when needed, in the next section of this chapter. 

EXAMPLE. By inspection, determine the singularities of f(z) = 

pi pt pZ 

. 4 . Since , . . = 7 -. -, f(z) has simple poles, that is, 



poles of the first order, at z = i and z = i. The expression e* 
becomes infinite at z = oo . The function /(z) is continuous except 
at z = db i and oo . If f(z) has a branch point then, by the definition 



LAURENT'S EXPANSION 255 

of a branch point given later, it is possible to obtain a value of z for 
which f(z) is at least double- valued. This is not possible for 

e* 
/( 2 ) ==: o Consequently the singularities of /(z) are poles at 

2r + 1 

z = i and an essential singularity at z = oo . 

113. Laurent's Expansion. Suppose that the point a is a regular 
point or a pole or an isolated essential singularity of /(z), but that 
/(z) is analytic within the annular connex R between two concentric 
circles whose center is a and whose radii are r\ and ^2- Then it can 
be proved (See Vol. II, Chap IV.) that within the annular region R 



/GO = c "( z ~ <*)" ( 317 > 

ft- -oo 

where the constants C n are given by the formula 



and where C is a circle whose radius is r, (ri < r < ^2) and whose 
center is a. The series in Eq. (317) is called Laurent's expansion of 
/(z). One of the uses of this expansion is the investigation of the 
nature of the singular points of single-valued analytic functions. 

It is provable that this expansion, like Taylor's expansion, is 
unique, i.e., there is only one two-way series in positive and negative 
powers of (z a) with constant coefficients which represents /(z) in 
the region R. Consequently, if such a two-way series can be found 
by any method it is Laurent's expansion for the given function /(z). 
Because the C n are, in general, difficult to evaluate by the above 
formula, a Laurent series is usually obtained by the easy method of 
the illustrative example of this article. 

If a is a regular point of /(z), then C n = for n = 1, 2, 3, .... 
If a is a pole of order m then (317), as will appear from the illustrative 
example following, is of the form 



- a) n . 



But if the point a is an isolated essential singularity then the series 
(317), the two-way infinite series, holds where the C n for n = 1, 
2, 3, ... oo are not zero. This is the distinguishing charac- 
teristic (and a definition) of an essential singularity. For example, 



256 PRELIMINARY MATHEMATICAL PRINCIPLES 

consider the function e 1/z . From 



00 





we obtain 

44 00 



2z n _ -s 
- (-)!~ 



(z 0)" 



The above series is the Laurent expansion for e l / z about the point 
a = 0, and because it contains powers of (z 0) infinitely large and 
negative, e l/l has an essential singularity at the point 2=0. 

EXAMPLE. It is desired to expand/ (2) = - - in a Laurent 

2^(2 + 2) 

series about the point 2 = 0, and to determine the nature of the 
singularity there. We may write 



2 3 2 5 

sin 2 = 2 +-- ... for all finite values of 2, 



The last expansion is obtained by the binomial theorem. We may 
multiply these series, provided z< 2. Finally then, 

sin 2 = _L _ 1 J: 



The function/ (2) thus has a pole of the first order at 2 = 0. 

114. Residues and the Residue Theorem. Let a be a pole of order 
m, or an isolated singular point, of f(z). Then the residue of f(z) at a 

is defined to be the coefficient, C-i, of in the Laurent expan- 

z a 

sion. But 



RESIDUES AND THE RESIDUE THEOREM 257 

Hence an equivalent definition of a residue is the line integral, 



where R denotes the residue, and C encloses no poles except a. 

Let f(z) have a finite number of poles or isolated singularities 
a n (n 1, 2, 3, ... m) in a connex R which is bounded by a closed 
curve G. For simplicity, consider m to be 3; then we have the three 
singularities ai, a2, and 0,3. Since /(z) is analytic except at #1, a2, and 
03, we have from Cauchy's theorem 

/ *f(z)dz + ff(z)dz + ff(z)dz + f f(z)dz - 0, 
JQ Jc\ */C2 J c$ 

where Ci, C2, and Ca are small circles enclosing a\, 0,2, and #3, respec- 
tively, and if the directions of integration are as shown in Fig. 67. 

The complete path is shown in the 
figure. The net integral along each 
cut vanishes as the two sides of the 
cut approach coincidence. The posi- 
tive direction of integration, however, 
is defined as such a direction that the 
area enclosed by the contour lies to FIG. 67. 

the left of an observer on the contour 

and facing in the positive direction of integration. Hence, as written 
before, the integrals about Ci, 2, Cs were taken in the negative direc- 
tion. If we write the integrals as taken in the positive direction, 




ff(z)dz = //()& + ff(z)dz + ff(z)d*. 
JQ Jci Jc 2 Jet 



But, by the second definition of a residue, 



f(z)dz = 2viRi, 
'i 

where RI is the residue otf(z) at a\. Likewise, in general, 



and 



/ f(z)dz - 2TriRn (n = 1, 2, 3, . . , m) 

*/Cn 



/ 
f(z)dz = 



258 



PRELIMINARY MATHEMATICAL PRINCIPLES 



which is the residue therem. One of the chief uses of the residue 
theorem is the evaluation of integrals in the complex plane. 

EXAMPLE. Evaluate the integral 

dx 



L 



+ !)' 




FIG. 68. 



This particular integral is more easily evaluated by integration along 

the #-axis, but we evaluate it here by integration in the plane in order 

to illustrate the use of the residue 
theorem. Although this is an integral 
involving a real variable only, it 
can be transformed into a contour 
integral in the complex plane. Evi- 
~ x dently the given integral may be 
replaced by one in the complex vari- 
able z, provided the limits remain the 

same, and the path be the #-axis. We consider the related contour (or 

line) integral. 

dz 

(z 2 + I) 3 ' 

RRAB(~R) 

where the path is the semicircle RRAB(-R) whose diameter is 
-RR. (See Fig. 68.) 
By the residue theorem 



dz 



-RRAB(-1 



where the number of residues is m, one residue for each pole. 
poles occur at the zeros of the denominator, at 



The 



z 2 + 1 = or 



- 1, z = 



The path in the above figure includes but one pole, namely z = i. 
Let us determine the residue at this pole from the Laurent expansion 
of the integrand about z = i\ i.e., we expand the integrand as a power 
series in (z i). To do this, write 



J_ 

+ 1) 3 



J_ 

(z- 



1 



RESIDUES AND THE RESIDUE THEOREM 259 

and expand - rr in a Taylor's series of the form 
(2 + *) 3 



. . . 
By this formula 

( _ + <)<*>- -0- 






Hence 

1 (z + )- 3 +* 3 3t 



(Z 2 + I) 3 (Z - i) 3 8(2 - *)3 16(2 ~ iY 16(2 - i) ^ 

Thus the residue is -, and 

16 



f 

.-. ( 



dz =7 ( - - 

" - - 



+ 

But also, by reference to Fig. 68, 

r dz r dz r dz 

-RRAB(-R) ~ R ^ ' R.A.B.-R 

Now let the radius of the semicircle R approach oo . As R oo the 

/dz 
, 2 t 1XH > 0, since if 2 be set equal to Re ie we have 



(2 2 + I) 3 

RAB(-R) 



i r w e ie de 



which surely approaches zero as R oo . Finally, 

dx 



7.00 



I) 3 ^.^(^ + 1) 



EXERCISES 



1. By reference to the Laurent expansion, prove the residue rules: 

(a) If a is a pole of order r of /(z), the residue of f(z) atz = a is the coefficient 
of (z a) 7 "" 1 in the development of the product (z d) r f(z). 

(b) If a is a simple pole of f(z), the residue of /(z) at z = a is equal to 

lim (z a)/(z). 



260 



(c) If r (z) 



PRELIMINARY MATHEMATICAL PRINCIPLES 
P(z) 



where both P(z) and Q(z) are regular at z a, and P(a) = 



and /(z) has a simple pole at z = a, then the residue of /(z) is equal to 
where Q' denotes the derivative. 

2 - x + 2) , 



/OO \ps _ 
_ . 
-*(x 2 -f- 1; 

3. Considering J , 



** + 9) ^ 

i around the contour show that 



/ oo cos a* IT . 2*- ,, ^=- /a A 

..rTr^-i-^ + T* C0i v~6/' 



where a > 0. 

(On the small semicircle let z -f- 1 = re i& , and on the large semicircle z = Re ie . 
Then let r -> and R -> oo .) 




^ 



FIG. 68a. 



FIG. 686. 



-' 






J2' 




1 






b 








1 




--R ^ 


FIG. 68c. 



_^_ 

1 +2 2 



4. Expand - in a Laurent expansion about the pole z = i. 



c cu ,u , r" giax s * e ~ ab ( b real and > > 

5. Show that / - ax = - , x, . ~. 

J - oo 2 _|_ X 2 ft ' (1 > a real > 0). 



Hint: Use contour shown. 
6. From Ex. 5 prove that 



cosaxdx ire' 



/- 2 _J_2 J ' 

/*> sin axdx 
7. Show that J e~ x * cos 2 Jx^x = -\/Tre~^. 

/oo 
e~ x ajc 
00 



HEAVISIDE'S CIRCUIT PROBLEMS 



261 



RIGOROUS AND SYSTEMATIC PROOFS OF HEAVISIDE'S RESULTS 

Heaviside's operational results consist mainly of the three well- 
known Heaviside rules. We are now in a position to establish rigor- 
ously these rules. 

115. Heaviside's Circuit Problems. Heaviside's circuit problems 
may be grouped under two headings: (a) " Unextended Problems," 
and (6) " Extended Problems." His unextended problem is as fol- 
lows : Given a linear network of n meshes in a state of equilibrium (no 
currents flowing or charges existing) ; find its response when a " unit " 
electromotive force is applied in any mesh. His " unit " function, 
usually written 1 or 1, is defined to be a 
voltage which is zero for / < and unity 
for / >: 0. 

The mathematical statement of 
Heaviside's unextended problem is : Given 
the n simultaneous linear differential """"" 
equations with constant coefficients which p IG 69 

specify the performance of an n-mesh 

network', find the response of this network when a unit electromotive 
force appears in one mesh. That is, given 



fit) 



Z\\i\ + 122 + . . . + Zinin = 1, 
Z2lil + 222*2 + . . . + Z 2n in = 0, 



il + Z n2 i2 + . . . + Z nn i n = 0, 



(318) 



find the solution subject to the condition that the network was initially in 
a state of equilibrium. In these equations 



Z rs = Lrsp + 



1 



C r8 p 



is the generalized impedance of the rth mesh to the sth current, i t \ 
L ri1 Rrs, and C T9 are constants, and 



. di. 1 . (* 

~* 9 y-~l ** 



where q, is the charge that has flowed in the 5th mesh. Methods of 
writing circuit Eqs. (318) for a given network are found in 20. 



263 PROOFS OF HEAVISIDE'S RESULTS 

Eqs. (318) are adequate for all circuits having lumped parameters L, R, 
and C for which Kirchhoff's laws are adequate. (See 20.) 

The current produced in mesh 1 due to the unit e.m.f. being applied 
in that mesh is called the indicial admittance, and is denoted by 
A(t). The current 4 produced in the &th mesh due to the unit e.m.f. 
being applied in the jth mesh (j different from k) is called the transfer 
indicial admittance, and is denoted by -4, *(/). 

The "extended Heaviside problem 11 differs from the unextended in 
that there is at least one applied electromotive force which is a variable 
function of the time. Stated mathematically, given the differential 
equations 



(319) 



where at least onef(f) is neither zero nor the unit function, find the solution 
subject to the condition that the network was initially in a state of 
equilibrium. 

Heaviside obtained three rules for the solution of the unextended 
problem which we will give later. He was not particularly concerned 
with the proof of these rules, but was primarily interested in their 
application. (The proof of 116 although original was suggested by 
Bromwich's proof, Ref. 39.) 

116. The Solution of Heaviside's Dnextended Problem. The 
solution of the system of differential equations (318) satisfying the 
initial conditions of equilibrium 



+ 222*2 + + 



*i(0) = 2 (0) = . . . = i n (0) = 0, 
= 32(0) = ...== fl .(0) = 0, 



(320) 
qi(0) = g 2 (UJ = . . . = q n (0) = 0, 

is 

1 r^(\)e^d\ 
ik = $(P)1 T : / (& = 1, 2, 3, . . . , n), (321) 

where 

* ( ^ = ^ 

and C is either a closed curve enclosing all the roots of XZ)(X) = 0, or 
a line from i<x> to +ioo of such form that all singularities of the 
integrand be on the left. D(p) is the determinant of the coefficients 



SOLUTION OF HEAVISIDE'S UNEXTENDED PROBLEM 263 

of i\ t*2, . . . , i n in the system (318), and M ik (p) is the cofactor of Z\ k 
in D(p). (For definition of cofactor, see 28.) 

We will now obtain the solution (321). A relation of the form 

(k = 1, 2, . . . , n) (t 0) (322) 

is desired which satisfies (318) and (320). If (322) for k = 1 is sub- 
stituted in the first term of the first equation of (318) we have in detail 

. . . / i \ i r K 

Zii\p)ii := I Lup -f~ jRn "f" ~ ) ~ I fi(X)g d\ 






where Xn = 2n(X). Similarly, the substitution of (322) throughout 
Eqs. (318) gives 






+ ... 



_/TW 

/ I r* \ 
Jc L OUA 



C lB X 



f [[X21&00 + . . + x 2n 

/ -^-^ + . . . + ^-^ \d\ = (323) 

f 

Jc 

/T^m A.ml 

x = o, 



-CnlX 

where 

X r . = Z W (X) (r, 5 = 1, 2, . . . , n). 
Since 



1+X/+ ^ 

2wi 



/\i ^ i -T A* T -T7- T /- , 

^ , / ^1 / aA 

d\=f dX = / 

_ X */^ X JQ X 



264 PROOFS OF HEAVISIDE'S RESULTS 

(See examples 2 and 3, 105), Eqs. (323) are evidently satisfied if 

+ . . . + Xln?n(X) I ' r 



... + 



+ ... 



Xnnn(X) - 



Ll 



+ . ..+ 



C nn X 



= 0. 



Although the last equations have no useful solution in the (X) yet 
they suggest a solution of Eqs. (323). Neglecting the coefficients 
of e~** and recalling that for / ^ the unit function is the real number 
one we replace tentatively the last equations by Eqs. (324). 



Xi 2 fe(X) 



= 0, 



X2fe(X) 



= 0, 



(324) 



If (324) are solved for & the value so obtained, owing to properties 
of line integrals, also satisfies Eqs. (323). If the determinant 

Xn Xi2 . . . Xin 

X21 X22 X2n 



Xnl X n 2 

is not zero, the solution of (324) is 



(325) 



where Afi^(X) is the co factor of Xu in -O(X). The substitution of (325) 
in (322) gives (321). 

It is now necessary to show that the solution (325) reduces the 
value of the second line integral in each equation of the system (323) 
to zero. Substitute (325) into the second integral of the #th equation 
of (323) to obtain 



(g=l,2,...,n). (326) 



SOLUTION OF HEAVISIDE'S UNEXTENDED PROBLEM 265 
A typical term of the above integral may be written 



f M lm (\) _ r 
Jc C tm \*D(\) J c 



X"Mi m (X) 



Now X X ra is a quadratic function of X, whence X n Mu is a polynomial 
of degree 2n 1 in X, and X n + 2 Z>(X) is a polynomial of degree 2n + 2 
in X. Hence, the integrand of the integral may be written 



+ 



+ . . . + a 2 n-i 



X 3 



J_ 

X 3 



00 + x 



The last fraction may be expanded in a Maclaurin series in powers of 
1/X, and the integrand may be written 



C 2 



\ 
'"/' 



The above series converges provided X is sufficiently large. Since the 
path of integration C encloses all the singularities, we may take it to 
be a circle of such great radius that this series converges at all points 
on it. By 105, example 3, 



Consequently, the integral (326) is zero, and it follows that (325) is 
the solution of (323). 

It remains to show that the initial conditions are satisfied. From 
(321), when / = 0, 

M n (\)d\ 



and 



X 2 D(X) ' 



Both these integrals may be shown to be zero by the method in reduc- 
ing (326), hence the initial conditions are satisfied, and the proof is 
complete. 



266 



PROOFS OF HEAVISIDE'S RESULTS 



117. Operational Formulas. If Eqs. (318) are solved for 
ik(k = 1, 2, . . . , ft), treating p as a mere algebraic symbol, we obtaL. 



Now it is known from the theory of differential equations that (318) 
has, subject to the boundary conditions (320), precisely one solution. 
This solution is given by Bromwich's integral 



2wi J c 



\D(\) 

The integrand of this integral can, of course, be transformed in many 
ways: resolved into partial fractions, expanded in series, etc. Con- 



sequently, the operator expression <> (p) 1 = 



D(p) 



1 may be simi- 



VYVVVYVVVVVVVVV- 

R 


k 

o 




o 




Lo 





o 







\\ c 


r 


II 





larly transformed as if p were merely an algebraic symbol. This is a 
result of great importance. There exist infinitely many algebraic 

operational formulas $ (p) 1 which can 
be evaluated by Bromwich's integral. 
Two of these are given in the illus- 
trative examples of this article, and 
many more are given in the exercises 
of this chapter. 

FIG. 70. EXAMPLE 1. Find the response of 

the simple series circuit, Fig. 70, when 

unit voltage is applied, the previous current and condenser charge 
each being zero. Take R = 12 ohms, L = 1 henry, C = 0.01 farad 
The differential equation is found by the method of 20 to be 



or 



Whence 



(1.0/> 2 + 12.0/> + 100) . 
P 4l 



P 



P 2 + np + 100 

Then from (321), 

. = j_ r 

1 2iriJ c 



1 = *(P)1. 



r c X 2 4- 12X + 100' 



HEAVISIDE'S EXPANSION THEOREM 



267 



The integrand has poles at X = 6 + iS and X = 6 i8, whence 
by the residue theorem and Ex. Ib of 114. 



X + 6 - iS 



x--6-8 



and the current is a damped sinusoid. 
EXAMPLE 2. The evaluation of 

, _ P 2 



may be accomplished by means of (321) as follows: 



, ._L f 

2V e 






\e* 



,d\. 



2-iriJf (X + ico) (X iw) 
The poles occur at X = iu, whence the integral is 



n 



X + ICO 



X ico 



e iut e- 



= ~2 + 2 



= cos< 



118. Heaviside's Expansion Theorem. Heaviside's first rule or 
expansion theorem gives the indicial admittance for a network of n 
meshes in the form 



AW - 



2n 

V. 



(327) 



where Xi, X2, . . . X2 n are the roots of D(\) = (all X's being distinct), 
and b is the coefficient of X 2n in J9(X), i.e., D(\) = b(\ - Xi) (X - X 2 ) 
... (X X 2n ). The denominator of the summation contains all the 
factors X* X which are not zero. 



268 PROOFS OF HEAVISIDE'S RESULTS 

The proof of (327) is accomplished from the Bromwich integral 
(321) by inspection. That is, if the residues of the line integral are 
calculated and added, formula (327) results. Since the process of cal- 
culating residues at simple poles is easy to remember whereas formula 
(327) is remembered with difficulty, the expansion theorem is seldom 
used. It is given, in this treatment, only for completeness. 

If D(\) = has zero or multiple roots, no difficulty arises. It is 
necessary only to evaluate the residues at multiple poles as indicated 
in Ex. 1 at the end of the first section of this chapter. The roots of 
D(X) = are found by GraefTe's method ( 46). 

Heaviside's second rule is: If $(p) can be expanded in a convergent 
series 



p 2 p* 
then 

a-/ 2 a / 3 

$(/>) i = a + ait + ~- + -V + (328) 

2 ! 3 ! 

The proof of this rule follows at once from the Bromwich integral. If 
the indicial admittance is desired for small values of the time (328) 
may be satisfactory. However, in general, (328) is used only as a last 
resort because the properties of the solution are frequently lost when 
expressed in series form. 

119. Fractional Powers of p. In the application of Heaviside's 
third rule, 120, to transmission lines and to heat-flow problems, 
fractional powers of p operating on unit function are encountered. 
But to evaluate p n l(() < n < 1) by the Bromwich integral it is first 
necessary to understand branch points of multiple- valued analytic 
functions. 

Consider the function W 2 = z, which may be written 

W = z* = + Vr (cos Y~ + i sin ^-j, 

where k = or 1. It is possible to consider \/r (cos - + i sin -) 

\ / 

as two distinct functions. Such 



. /-/ + 27T . . + 27T\ 

and v r I cos - -- r * sin - ) 



is not done, but they are considered as two branches W\ and W 2 of the 
one function W. We are thus led to define a branch of a function. 
Let there be a set of pair values (W, 2), such that all the z's considered 
fill exactly once a region R. Let there be a one-to-one correspondence 



FRACTIONAL POWERS OF p 



269 



between the points of the region R and a region in the W-plane. The 
points in the PF-plane defined by the one-to-one correspondence form 
a continuous function of z and are called a branch of the function W. 
In the example W 2 = z, let z traverse in the 0-plane any path ABCDE 
which encircles the origin. (This path is the region occupied by z.) 




W-plant 




FIG. 71. 



For simplicity, let the path be a circle whose radius is r (Fig. 71). Then 
z = re ie . As 6 varies from to 2ir, \fr (cos - + i sin ~ J = \/r e i(e/2} 
describes a semicircle A'B'C'D'E' of radius \/r, in the upper half of the 









) = \/r 



r e (0+ ' 2ir)/2 describes 



TF-plane, while -\A( cos -- h i sin 

a semicircle A"1$"C"D"E" of the same radius in the lower half of the 
W-plane. When 6 (in the z-plane) takes on the value 2?r, the branches 
of W interchange. That is, if a point PI traversed the branch A'C'E 1 
while the point P> traversed the branch A" C" 1L" as increased from 
6 to 2?r, then as 6 increases from 2ir to 4?r, P% traverses A'C'E! , and PI 
Ira verses A"C"E". 

In general, a point a is a branch point of the function W = f(z) 
if the branches of W interchange as z encirles a. In the example 






\C 2 



W-plane 



z- plane 



FIG. 72. 



W 2 = z, the branch point is z = 0. Suppose z traverses a curve C in 
the z-plane which does not encircle the origin. (See Fig. 72.) Then, 
as <j> varies from to 2?r, Wi and W* describe the two closed curves 
Ci and 2. As <f> increases from 2ir to 47r, each of the branches is 



270 PROOFS OF HEAVISIDE'S RESULTS 

described again, but the point PI on C\ always remains on Ci and the 
branches are never interchanged. Hence z = a is not a branch point. 

Obviously W = log z = log | z \ + i arctan ( 2irni has an infinitude 

x 

of branches, and z = is its branch point. 

It was pointed out in 112 that branch points are singular points 
of multiple-valued functions. The theorems of 108-114 do not 
hold when branch points are enclosed by the contour of integration. 
This raises the question of the treatment and geometrical representa- 
tion of multiple-valued functions of a complex variable. In the case 
of real functions, such as y =t \/x, both branches of the function are 
drawn in the :ry-plane for real values of x taken along the positive 
x-axis. In the case of multiple-valued complex functions the W-plane 
is made up of more than one plane or sheet; one sheet for each branch. 
These sheets are joined at the branch point. For instance, in the case 
of W = log z the branch point is z = and the entire W-plane is rep- 
resented by a helical surface such as a winding stairway which winds 
about the point z = 0. That is, since 

W = log p + id, 

as z describes a circle about z the value of p remains constant but 
id increases by 2-rri with each revolution. If a plane, containing the 
axis of the spiral, extends outward on one side from this axis, it cuts 
the spiral surface into infinitely many sheets. If a contour is drawn 
on any one of these sheets in such a way that it does not cross the 
curve formed by the intersection of the plane and the spiral surface, 
then log z, for the values of z on the contour, is a single- valued analytic 
function and the theorems so far proved apply on any single sheet. 

The value of p n l(\ > n > 0) is frequently required. It has been 
shown in 116 that (321) is the solution of the circuit equations for 
<(/>) an algebraic function of p. It has been shown elsewhere (see 
Ref. 43 at end of text) that (321) is the solution of the circuit equation 
for $>(/>) an analytic function. Accordingly, by the Bromwich integral 
(321), 



= J__ / X-V'dA, 
27TZ./<7 



where C is the line described in exercise 13 122. 

The integrand evidently has a branch point at X = 0. However, 
on and within the contour shown in Fig. 73a the integrand is single- 
valued and analytic. 

In most physical systems the real parts of the roots are negative, 
hence there are no singularities of the integrand in the right half of 



FRACTIONAL POWERS OF p 



271 



the z-plane, and contour (a) is replaceable by (b) in Fig. 73. Applying 
Cauchy's theorem to the contour (b) we have 



Origin 





Ki(i. 73. Contour for Fractional Exponent. 



(i) 



+ f 

/;/2)- 

f (pe ir ) 
JR 

f \re 

/r 

/ 
(pc-' 
, 

/-(ir/2)- 
j- 



_ sn 



cos 



(ID 



*~ l e tpcosir e iir dp 



(in) 



~ sn 



cos 



(IV) 



(V) 



+ 

= 0. 



cos 



(VI) 



The limits on integrals III and V are positive quantities since the 
transformation on X is X = pe i9 . When X = R + Oi and = TT, then 
p = R. As jR > oo and r > 0, it is easily shown that integrals II, IV, 
and VI are zero. (See Ex. 11 at the end of this section.) Hence 



1 /* 
- I 

"Jo 



sin (it - l)ve~ tp dp 



sn 



Let pi = y. Then 



272 PROOFS OF IlKAVISIDE'S RESULTS 

The last integral is a gamma function T(n). It cannot be expressed 
in terms of a finite number of elementary (calculus) functions. But its 
value for all real values of n can be found in a table of gamma functions. 
Finally then 

pni = ^LUm: r(w ) (i > n > o). (329) 

7T/ n 



Evidently 

. d [(sin tt7r 



)r(n)"| 

\ 

(330) 



n(sin nw)T(n) 



In a similar way, /> Mfr jf (r a positive integer) can be written down at 
once. 

120. Heaviside's Third Rule. Heaviside's third rule is: If 
can be written as 

*() = (00 + ai + 02/> 2 + ...)+ P 2 (bo + bip + bop 2 + . . 
then 



for / > 0. There has been much controversy over the validity of this 
rule. To see that it is true, it is necessary only to substitute Q(p) in 
(321) and make use of (329), (330), and a table of gamma functions. 

121. Heaviside's Extended Problem. Thus far we have been con- 
concerned only with Heaviside's unextended problem. The extended 
problem stated mathematically is: Given the n simultaneous linear 
differential equations (319} with constant coefficients where the voltages 
/t(0 (^ ' I* 2, . . . n) arc suddenly applied to the network at t = 0; 
one voltage in each mesh. At least one /() is neither zero nor unit 
function. No charges exist and no currents are flowing at time t = 0. 
Find the response of the network. 

The procedure is as follows: First solve Eqs. (319) under the 
restriction that /,(/) = 0(* = 2, 3, . . . n) and fi(t) ^ 0. Next let 
fi(f) = 0(i = 1, 3, . . . n) and/2(0 5^ 0. Carry on this process, finally, 
solving (319) for all/t(/) = except /(/) Now it is known from the 
theory of such differential equations that if the n values obtained for 
(say) 4 are added, the total value for i* so obtained is identical to that 
obtained by solving (319) with no/ t -(0 = 0. This is the well-known 
principle of superposition. Consequently it is necessary only to 
investigate the operational solution of 



HEAVISIDE'S EXTENDED PROBLEM 273 



= 0, 



(332) 

0, 

subject to the initial conditions of equilibrium configuration. The 
symbol f\(t) 1 signifies that the voltage f\(t) is suddenly applied at 
t = 0, the previous voltage being zero. 

Suppose that the indicial admittance A(t) has been found; that is, 
Eqs. (318) have been solved subject to the initial conditions of the 
preceding paragraph. Then it may be shown that the current in the 
first mesh due to the suddenly applied voltage 1 /(/) in that mesh is 



- f 

Jo 



A(t- X)/'(X)dX. (333) 

This is Duhamel's superposition integral. In this formula /'(X) means 

- with / then replaced by the real variable X. Before proving (333) 
at 

let us illustrate its use. 

EXAMPLE 1. A circuit consisting of an inductance L and condenser 
C in series is connected at / = to an alternator supplying voltage E 
cos co/. Find the current. The indicial admittance is 

A(f) - ! 



1 . 

= sin at, 
aL 

where 

--re 

Then 

f(t) = E cos w/, /'(X) = &E sin coX, 
and 

Ti /** 1 

i(t) = sin at + / sin a(t X) ( toE) si 
aL ,/ aL 

E coE f* 

= sin at + v / cos (wX aX + 
aL 2aLy 



274 PROOFS OF HEAVISIDE'S RESULTS 



coE (* 
2aLj 



cos (wX + aX at)d\ 

E . , w [ sin ut sin a/ sin w/ + sin a/1 

= sin at + ~~ 

aL 2aL L a> a w+a J 

TV 77 1 

= sin at + ^ ~ (a sin <o/ co sin a/). 

fljL " 




Relation (333) is derived by adding or superposing the currents 
due to each increment of voltage f(t) from / = to / = /. We are 
interested in the voltage at time /. If the voltage E is applied at time 

t = 0, the response is EA(). If 
another voltage Ei is applied at 
time / = X the new portion of re- 
sponse is E\A(t X). The total cur- 

t rent then at time / subsequent to 

FlG 74 / = X is EA(f) + EiA(t - X). Con- 

sider now the voltage f(t) represented 
by dotted curve in Fig. 74. Let this voltage be replaced temporarily 
by the stair-step voltage drawn in the solid line. The increment of 

voltage at / = X then is AX. The current due to this increment 

dt 

is A(t X)/'(X)AX. The current due to the sum of such increments is 

A(t- 



As AX > and n > oo the stair-step voltage approaches /(/) and the 
sum becomes 



There must be added to this current the current due to the voltage 
/(O) applied at / = 0. Thus the total current is 



f* 

JQ 



A(t - X)/'(X)dX, 



and (333) is established. Eq. (333) gives the total response, that is, 
transient plus steady-state current. If the current in mesh k due to 
impressed voltage in mesh j is desired then we have 



i k =/(OM*(/) + A Jk (t - x)/'(x)dx, 

A 
where A* is the transfer indicial admittance defined in 115. 



SUMMARY 



275 



EXAMPLE 2. Consider the circuit shown in Fig. 75. Write the 
operational expression for the transfer indicial admittance. 



I vwwv\ i 



Rt 



-*j r r 



Cj (a) (6) 

FIG. 75. Transformer Circuit. 



The differential equations are (if positive currents produce opposing 
fluxes) 

[Lip + RI + 
\ Cii 

( Y 

V ~ Czp/ " 



t2 



+ *i + ~- 1 
C\p 

- Mp 



Lip 



1 



Mp 



L 2 p + R2 + 



If 4>(p) is substituted in (321), Aw(t) is readily found as a function of 
the time /. 

122. Summary. If only one voltage is impressed suddenly on a net- 
work the steps of the solution are as follows: 

(a) Write the differential equations of the network by the princi- 
ples of 19 supposing the applied voltage to be IE. 

(b) Solve for the current (say i/t) by determinants, obtaining 



(c) Evaluate this operational expression by substituting in the 
Bromwich integral (321). If IE was applied in the fcth mesh, then we 
have the indicial admittance. If IE is applied in some other mesh, the 
above expression is the transfer indicial admittance. 



276 PROOFS OF HEAVISIDE'S RESULTS 

(d) Now use the superposition theorem (333) where the voltage 



(e) If n voltages are applied, one in each mesh, repeat steps (a) 
to (d) for each voltage and add the resulting currents. 

This short section thus contains the elements of the Heaviside 
operational calculus. 

EXERCISES AND PROBLEMS 

Establish, by means of the Bromwich line integral, the ten operational formulas 
following 

1 / m 

1. ~ M 1 (m a positive integer). 

2 _ P __ / - /,-w 
<# + '' ' 

3. -- P - I = -1-k-w - e ~) m 

^ ' 



4 - 



a. -*-- 



(p 2 + a 2 ) (/> + tan /3) 

s. (a) p^ ^ - OT/) 



^ [a)S 



10. , f ^^ 6r sinu,/. 
(/> -f W 2 4- w 2 

11. Show that integrals II, IV, and VI of 119 are zero. 

12. A voltage is suddenly impressed upon the primary circuit of a transformer. 
Find the currents i\ and i* at any time thereafter when E 100 volts, L\ = 1.0 
henry, L 2 = 2.0 henrys, M - 0.5 henry, RI = 10.0 ohms, and R* = 5.0 ohms. 

13. Establish the Bromwich integral Eq. (321), 116, where the contour is a line 
from i'oo to H i'oo such that all singularities of the integrand lie to the left of this 
line. 

14. Repeat the reasoning of 116 if every z r * in Eqs. (318) is replaced by z? s , 
where m is a positive integer. 

15. Repeat the reasoning of 116 if the unit function is defined to be a voltage 
which is zero for t < 0, equal to every value from to 1 inclusive at t = and equal 
to unity for / > 0. 



ALGEBRAIC OPERATIONS AND SHIFTING 277 



III 

MANIPULATIVE DEVICES AND APPLICATIONS OF THE OPERATIONAL 

CALCULUS 

The last section contains all the essentials of the operational cal- 
culus. The fact that (321) is valid for <>() any analytic function is 
a result of great importance. However, there exist certain tricks or 
devices which frequently lessen the labor of obtaining results. It is 
the purpose of this division to explain some of the most important 
of these and to illustrate applications of the operational calculus in 
the analysis of transmission lines, the study of heat flow in refrigerator 
walls, and the design of brake shoes for the rotors of large gen- 
erators. 

123. Algebraic Operations and Shifting. Although the Bromwich 
integral always gives the correct answer to the problem, it is sometimes 
convenient to evaluate operational expressions by other methods. 
For example, the expression can often be simplified materially before 
being substituted in this integral; in some cases, it may even be 
simplified to such an extent that it can be recognized as a standard 
form. A number of standard forms or formulas are found in 122. 
Two simplifying processes of particular importance are the algebraic 
transformations of the operator and the so-called " shifting " processes. 
These are now considered. 

In the development of the Bromwich integral, it was shown that 
it is legitimate to perform any algebraic operations on 3>(p) that are 
valid for the integrand of (321). Two such operations that are fre- 
quently useful are expansion into series and decomposition into partial 
fractions. Examples of series expansions were given in Heaviside's 
second and third rules above. 

EXAMPLE 1. Evaluate -I by the method of partial frac- 
tions. Since in the integrand of the Bromwich integral (321) 



X 2 + a 2 2 \X + ia X - ia, 
we may write 



(P + w)(p - ia) 



1 + 



278 APPLICATIONS OF OPERATIONAL CALCULUS 

The last two fractions are immediately evaluated by means of the 
formulas of 122. Whence 



= (1 _ e -i/) - (1 - 

2Ua ' *a 



sin at. 



2ia a 

By means of the " shifting " process, a factor may be removed 
from an operational expression before evaluation. If the presence of 
a certain factor in the final result is suspected, it is natural to suppose 
that the removal of this factor will simplify the evaluation of the 
operational expression which remains. 

In the Bromwich integral (321), multiply by 6< and obtain 



In order to integrate this expression, change the variable from X to 
X', where X = X' a. Then 



2wi J c X' 



Therefore, 



This formula shows the form taken by <(/>) when an exponential 
e~ at is factored out of the final result. 

EXAMPLE 2. Evaluate ------- - /. Here, owing to the presence 

(p + a) 

of the quantity p + a, we suspect e~ at as a factor. The removal of 
this factor gives 



P n 



ALGEBRAIC OPERATIONS AND SHIFTING 279 

Finally, by Ex. 1, 122, we have 

$(p) 1 = e~ at (for n 1 a positive integer). 

(n 1)! 

Example 3. Evaluate again ~ 1. The denominator may 

be written (p + ia)(p ia), where we may suspect the presence of a 
factor e** '. Its removal gives 



(/> + * 



[. . ~t 

p-ia'p(p- 2ia) J 



= - sin 



p - 2ia 

^ 1 _ e 2iat 

2ia a 

from Ex. 5, 122. In this example, the final result did not contain 
the suspected factor e~ iat , but its removal simplified the solution. 
By means of the Bromwich integral we may examine the effects 

of factors of <b(p) vsuch as p + a and 

p + a 



2iriJ(; 

^ /" 

2iri J c 

2iri dtJ c 



at 
In particular, if a = 0, 



thus identifying /> as a differentiating operator. If a is not zero, then 
by performing the indicated differentiation, 

(P + o)*(p)1 = ~ 

at 



280 APPLICATIONS OF OPERATIONAL CALCULUS 

thus illustrating again the fact stated previously, that algebraic open 
ations on <() are permissible. 
Again, 



p + a 2iri c (\ + a)X 



rn 

J-vllri 



X + a 
since 



- f e*dt. 

J - oo 



X 
Change the variable from X to X', where X = X' a. Then 



P + a 

r l 

- a) /] dt. 



Since $(p a)l is zero when / < 0, the integration from oo to 
therefore yields zero, and the result may be written 



In particular, if a -- 0, 



P J* 

thus identifying \/p as an integrating operator. 

EXAMPLE 4. Evaluate (p a)"' 3 1. By means of the formula 
just established, 

i i = p f = i f t2g _ atd( 

(p + a)* ~J C " p 2 ~ 2J 

= [1 - (1 + at + -| a 2 t 2 )e-<"]. 

ft 

124. The Derivation of the Partial Differential Equations of the 
Transmission Line. The differential equations, 19, for circuits with 



PARTIAL DIFFERENTIAL EQUATIONS 



281 



concentrated parameters (inductance, capacitance, and resistance) are 
ordinary differential equations. On the other hand, the equations for 
the transmission line current and voltage, since the parameters are 
distributed, are partial differential equations. These equations may 
be derived by applying Kirch- 
hoff's laws ( 19) to an ele- 
ment of length of the line. A 
single line with ground return 
is considered. 

Accordingly, referring to 
Fig. 76, the voltage above 
ground at A plus the rise 
of voltage in the infinitesimal 
length dx is equal to the voltage above ground at B. 

Let the notation be: 

e voltage at a distance x from the sending end of the line. 
i = current above ground at a distance x from the sending end 
of the line. 



y 


4 >t / 


? 




Hr -J 




Y . 

sorsin wt 






i 




'*fe* 



FIG. 76. Transmission Line. 



C capacitance 

R resistance 

L = inductance 

G = leakage conductance 



per unit length of one wire of the line. 



Then, if the voltage at A is e, at B, e -\ dx, and the rise of 

ox 

I di\ 

voltage is [Rdxi + Ldx--], 



e Ridx L -- dx = e H dx, 

At dx 

or 

9^ D ^ r di j 

dx = Ridx L-r dx. 

3x At 



In a similar way, by KirchhofFs second law, the current equation is 



dx = Gedx + C-r dx. 

dx At 



282 APPLICATIONS OF OPERATIONAL CALCULUS 

These equations evidently reduce to 



(334) 



- - (G + Cp)e t 



where p = . These are the transmission line equations. 
By differentiation and substitution, Eqs. (334) become 



dx 2 



= 0, 



(335) 



where n 2 = (R + Lp)(G + Cp). 
seen to be 



By 11, the solutions of (335) are 



(336) 



e = 



where the X's, which are either constants or functions of the time 
only, are determined by the conditions of the particular transmission 
line problem to be solved. 

125. Transmission Line Problems. Owing to various terminal 
conditions on transmission lines and the different assumptions made 
regarding the relative magnitude of the distributed constants, Eqs. 
(336) lead to a very large number of different transmission line prob- 
lems. From the operational point of view, these problems have cer- 
tain features in common. In general, the operator p is involved, both 
irrationally and transcendentally, in the operational expressions 
resulting from (336). The total current, both transient and steady- 
state, can be obtained from the indicial admittance by use of Duhamel's 
superposition theorem. Two particular problems are now considered 
illustrating these features. 

EXAMPLE 1. Infinite distortionless line. Let it be required to 
find the transmission line current at any point x and at any time t 
due to impressed unit or sinusoidal voltage, when the constants of the 
line are such that RC = GL. Such a transmission line is said to be 
distortionless. If the line is of infinite length, the current and voltage 



TRANSMISSION LINE PROBLEMS 283 

will both decay for x sufficiently large and hence K$ ~ K in 
Eqs. (336). Thus 

i = e- x Ki 
and 

so that, using (334), 

= - ne-**Ki = - (G + Cp)e = - (G + Cp)e- nx K 2 . 

Therefore, 

n 



When x = 0, i = Jfi, and it is seen that K\ is the current entering 
the line. Likewise, Ki is the impressed voltage. If the unit e.m.f. 
is impressed, the entering current io will be given by 



while the current at any point along the line is 



Cp 



+ LP 

and the voltage at any point is 

e = e~ nx 1. 
Since RC == GL 



and 



so that the entering current is steady. The current at a distance x 
down the line is 



fc e -**VcJl e -*VLd* 



284 APPLICATIONS OF OPERATIONAL CALCULUS 

The operator e ~ x ^ LCv 1 which occurs here may be evaluated by means 
of the Bromwich integral : 

e-"l = 

2iri^ c 

This is the form of the unit function with t a instead of /. Its 
value is therefore zero when / a < and unity when / a > 0. 
The solution of our problem is then 

i = for / < xVLC, 

-VORx or 



The current at any point is thus zero until the current " wave " has 
had time to reach the point, after which it remains at a constant 
value. The propagation velocity is 



By means of the superposition theorem, we may investigate the effect 
of impressing a voltage eo = E sin ut at the terminals. Here 

e(\) = E sin coX, e(G) = 0, 

A(f) =0 for / < x\/LC, 

A(t - X) = for X > / - x\/LC, 



for / > 

for \<t-xVLC. 



Then by (333) 

i(f) = c(0)A() + A(t- \)e'(\)d\ 
JQ 

/t-xVLC s*t 

A(t- \)e'(\)d\ + _A(t- X) 

St-l\/LC 

In the first of these integrals, A (t X) is a constant, while in the 
second it is zero. Hence 



TRANSMISSION LINE PROBLEMS 285 

This formula shows the form of the current due to any impressed 
voltage e(f). In particular, for the sinusoid, 



Thus, irrespective of the frequency the current wave is attenuated 
the same amount for a given distance x. It is for this reason that the 
line is called distortionless. 

EXAMPLE 2. Open-circuit ideal cable. Let it be required to find 
the voltage and current at time / at any point x of the open circuited 

ideal cable of Fig. 77. Essen- { 

tially, an ideal cable is a transmis- |~ TI 

sion line which has insignificant \ j 

leakage and inductance. i j 

In Fig. 77 the distance x is V [ x 

measured from the receiving end r ' '; l .,.,,,. \ ^m-rr 

. '/~/"/S//'//////7/////////////////'///////////S///'> 

of the line, instead of from the P ~- 

sending end as in Fig. 76. Con- 
sequently it is easily verified that all algebraic signs in the differential 
equations corresponding to Eqs. (334) are positive. Eqs. (335-336), 
however, are unchanged. 

From the general definitions of e* } sin 2, and cos z ( 109) and from 
the definitions of cosh x and sinh x given in the calculus, it readily 
follows that 



e -nx cosn nx sinh nx. 

Consequently the second of Eqs. (336) may be written 
e = (K+ Kz) cosh nx + (K Kz) sinh nx 



(337) 
= A\ cosh nx + A<2 sinh nx. \ 



Then by Eqs. (334) 



= (G + Cp}e = (G + Cp}(Ai cosh nx + A 2 sinh nx), 



or 

(C* -4- 



nx _|. ^ 2 cos \ l ny ^ (338) 

n 



286 
Since G 

and 



APPLICATIONS OF OPERATIONAL CALCULUS 
L = (ideal cable), 



n 



rf (^i sinh 

Jfv 



+ 4 2 cosh 



Let the applied voltage be 1 E. Since the line is open-circuited, it 
follows that 

i = for x = 0, 



e = IE 



for 



= /. 



Substituting these boundary conditions in Eqs. (337-338), it is obvious 
that 

A2 = 0, e = AI cosh nx, and :/ = A\ cosh w/. 
Finally, 



cosh 






_ (a)" **?,. 

\R/ coshnl 

Evaluating (340) by (321), we have 

. = (Cp\* sinh nx = __ / CY* /^X H si 

1 ~ \R) cosh n/ ~ 27T/ \^/ 7 C X cosh (l\/RC\) 



(340) 



sinh 



^ 

* ' 



27T/ C 

(where now j = V 1). Let the variable of integration be changed 
from X to z by X = z 2 . Then 



A 

Z ' 



. = E / C\ w f sinh 
" TT; \jf?/ ^ cosh 



cosh (lz\/RC) 

The variable z runs over the semicircle C\ (Fig. 78) in the upper half- 
plane as X traverses the infinite circle C. But 
\fj ^\ ; C\ is not a closed contour and consequently the 
residue theorem does not apply. Since there 
are no branch points of the integrand within 
the contour let C = 2 be a circle twice drawn. 
The value for i then becomes 




E 
- 

47r; 



./ c , x cosh (/V RC\) 



EQUATIONS OF LINEAR HEAT FLOW 



28) 



Let X = . The value for i then is 

xvC 



i = -*L f*l 

2irRJ^ c 



sin xu 



2TrRJ Ct cos lu 
where 3 is a circle enclosing the roots of cos lu. These roots are 

= ^f-^y (*-0,l,2,...). 

The residues of the integrand, by Ex. 1 of Sec. I of this chapter, are 
[ M _(2liLl)^ sir _-(/* 



R. 



Lim 

Jto-ly 

. (25- 



COS 



(5 



. (25 - l)r 
/ sin 



AC 



(5 = 0, 1, 2, . . . ) 



Finally by the residue theorem, 1 14, 



f 



;-i) 



,+i 



-I 




In a similar manner, it can be shown that 

e = 

126. Partial Differential Equations of 
Linear Heat Flow. It may be seen that 
the differential equations for the flow of 
heat in one direction are precisely Eqs. 
(335), provided L = C = 0, and e and i 
denote respectively temperature and flux 
of heat at the point A in Fig. 79. 



Let the parameters pertaining to the 



'// ' '/ Wall '/'//// 

'//////////<'///. 



'"/'//'"'/'/. 



flow be: 
R 
C 



FIG. 79. Refrigerator Wall. 



thermal resistance per unit length, 

heat capacity, that is, the heat absorbed by the material 
per unit length per unit increase in temperature. 



288 APPLICATIONS OF OPERATIONAL CALCULUS 

Let the temperature drop be taken positive in the direction of heat 
flow. Consider an infinitesimal length dx of the material. If the 

temperature at A is e, then the temperature at B is e H dx. The 



temperature drop across dx is Ridx. Hence, 
By similar reasoning 



~dx = Ridx. 



di , de , 
dx = C dx. 



The equations of linear heat flow are 

!*', 

(341) 



?-* 

a* 



or 

= 0, 



127. Refrigerator Box Heat Leakage. The transmission line the- 
ory so far developed is of great value in calculating refrigerator box 
heat leakage. Consider a particular case. To correlate calculations 
and test, an experimental box was built having the following specifica- 
tions. The inside dimensions were 5 by 5 by 4 ft. with cork walls 
having a thickness of 4 in. Both the inside and outside of the box 
were lined with Tpg-in. sheet steel. The fits between the inside lining 
and the cork filler and the outside lining were tight, so that no air 
spaces existed at any point between the two layers of the metal. The 
cork used in the box had a conductivity of 6 B.t.u. per sq. ft. of surface 
per in. of thickness per degree F. per 24 hr. Its density was 10 Ib. 
per cu. ft., and its specific heat was 0.485. 

One of the check tests was conducted as follows. An incandescent 
lamp rated at 100 watts was suspended at the center of the volume of 
the box. At the initial instant, the box was at the ambient tempera- 
ture 70 F. By use of thermocouples, data were obtained for a curve 
between the temperature of the inner lining of the box and the time, 



REFRIGERATOR BOX HEAT LEAKAGE 289 

measuring time from the instant that voltage was applied to the lamp. 
The outside of the box was 70 F. 

Let it be required to calculate a temperature-time curve for the 
inner lining of this box and plot the results up to the point at which the 
temperature reaches an approximately constant value. 

For a first approximate result, it is sufficient to consider the heat 
density to be applied uniformly over the inner wall of the box. The 
effect of corners is neglected. The metal linings are also neglected. 
Fig. 79 and Eqs. (341) are respectively the figure and differential 
equations of the problem. By the method of 125, the general solu- 
tion of the equations is 



6 = A\ cosh nx + A% sinh nx 

pC 

i = (A\ sinh nx + A<z cosh nx), 
n 



(342) 



where n = (RCp)^ and 6 is temperature difference of the point whose 
coordinate is x and the exterior wall. Evidently, from the conditions 
of the problem, = for x 0, and i = io for x = /; #o being the 
temperature of the exterior lining of the box, and io the flux of heat 
applied to the interior lining. 

Substituting these terminal conditions in (342), there results 

nio sinh nx 

H == -.1 

pC cosh nl 

This expression is evaluated by the Bromwich integral (321) as follows: 

1 C i 

1 / mo sinh nx x , 

4fajJ c \ 2 Ccoshnl 
where c is a circular contour twice drawn. 

As in 125 let X H = /T ^M or X = - . Then 

(_/ 

e-* l/RC du, 



J_ C Rip si 
2irjJ Ci u 2 co 



cos ul 

where c\ is a circular contour. The roots of the denominator within 
the contour evidently are u and 

2s + 1 v ft 
- 7T~ 7. *-0,__l, 2, 



290 



APPLICATIONS OF OPERATIONAL CALCULUS 



The residue at the double pole u = is Riox. The residues at the 
remaining poles within the contour are 



sin 



R. = Rio - 



/2s + i\ 2 ** 

\ 2 ) I 



1 /2J+1\T 

RC\ 2 ) fl c = n -4- 1 4- 2 

, J U, Ztl, -6| .... 



Finally, by the residue theorem of 114, 

:-i)' . 25 



Ri r + " 



(*i\ 

V 2 ) 



1 /2*+iy,rl 
/ZC\ 2 J ft' 



To obtain the temperature at the interior surface, set x = /. We 



The constants of the solution are: 

C = 0.485 |^r = 4.85 f |^- 

1 6 B.t.u. in. 6 B.t.u. ^ ft. 
./ = ~ 



= _ 1 B.t.u. 

ft. 2 day F. ~ ft. 2 86,400 sec. F. ~ 172,800 ft. sec. F. 

= 838,000 ~ 



ft ser F 

RC = 172,800 "** ^ X 4.85 



r>.t.U. 



sec / 4 \ 2 
RCP = 838,000 ^ X ( ) ft. 2 = 9430 sec. 

io(area) = 100 watts X 0.000949 B ' t ' U ' = 0.0949 B-t ' U ' 



watt sec. 
O.Og^B.t.u. per sec. _ ^^ B.t.u. 



sec. 



140 ft. 2 



ft. 2 sec. 



ft sec F B t u 1 

Riol = 172,800 - '-' X 0.000678 ^- X - ft. = 39.1F. 



B.t.u. ft. 



ft. 2 sec. 3 



Then the temperature difference 0(1) of the exterior and interior 
walls is w 

e(l) = 39.1F. - 7.92F. V( jVC^)^. 

fTo\ 2 * + 1 / 

The curve of ^ as a function of the time is shown in Fig. 80. 



WATER-WHEEL GENERATOR BRAKE 



291 



128. Water-wheel Generator Brake. Water-wheel generators are 
frequently equipped with friction brakes to stop the rotor and hold it 
against the torque due to leakage through the water gates. These 
brakes consist of a number of brake shoes disposed so as to apply 
pressure at a number of points around the horizontal surface of the 
rotor, rubbing on an annular steel plate, called the brake plate, bolted 
to the spider. The heat generated by the friction of the brake raises 
the temperature of the brake plate, setting up severe internal stresses 
which on several occasions have resulted in fractures. In designing 
such a brake, some questions which arise are: how thick should the 




4 B It 16 SO 24 28 32 39 40 

Time in Houn 

FIG. 80. Temperature Difference of Interior and Exterior Wall. 

brake plate be made, how much pressure should be applied on the brake 
shoes, and what are the maximum temperatures in the brake plate 
under various conditions? 

The machine to be considered here is a 68-poIe, 88.3-r.p.m., vertical 
water-wheel generator. The rotor consists of a spider built up of 
steel plates bolted together, with the field poles dovetailed around the 
periphery. The brake plate is bolted to the under side of the spider. 
It has a mean radius of 171 in. and a radial width of rubbing surface 
of 10 in. There are 20 brake shoes each 9 by 18 in. spaced equally 
around the plate and slightly staggered radially so as to completely 
utilize the 10-in. rubbing surface. Each shoe may be pushed upward 
against the brake plate by means of two hydraulic cylinders 6 in. in 
diameter. The coefficient of friction on the braking surface is 0.25. 

The brake shoes are made of an interwoven copper-mesh asbestos 
compound. The brake plate is made of steel with the following 
thermal properties: 

Conductivity = 0.46 watt per cm. 2 per C. per cm. 
Specific heat = 3.9 joules per cm. 3 per C. 



292 



APPLICATIONS OF OPERATIONAL CALCULUS 



The WR 2 of the rotor is 1.69 X 10 8 lb-ft. 2 The brake is applied 
at full speed, and the pressure in the operating cylinders remains 
constant. There is a gate-leakage torque of 200,000 lb-ft. which is 
approximately 3 per cent of full-load torque. The gate-leakage torque 
may be assumed not to vary with speed. 



68 poles 
equally spaced 



to shoes 
equally spaced 



Enlarged Section at AA 
Spaces for heads of spider through -bolts 
f+\l\+-3-fal #1-V 4 %"^ 





surface 



FIG. 81. Brake Plate and Shoes. 



EXAMPLE. Let it be required to determine the relation between 
maximum brake surface temperature and brake cylinder pressure. 

Solutions, of course, may be made based on various assumptions. 
Since the brake plate is about 2 in. thick and the heat is applied for 
about 2 minutes at most, we first consider the problem equivalent to 
an infinite transmission line problem. It is assumed that the loss of 
heat from the sides of the brake plate is negligible. 

The flux of heat or heat density applied is variable. The surface 
temperature, due to the application of this variable heat density, is 
calculated by means of the superposition theorem, Eq. (333), after the 
surface temperature due to a suddenly impressed constant heat density 
has been determined. 

Referring to Fig. 76, and the first equations of example 1, 125, 
we have 

i = er*K\ 
and 

e = e" K 2 , 

where, from the above assumptions, G = and n = \/RCp. Let io 1 
be the heat applied per unit area at time / = and eo be the surface 



WATER-WHEEL GENERATOR BRAKE 293 

temperature at x = 0. From Eqs. (334), = Cpe. Substituting 

the value of e above in this differential equation and integrating with 
respect to x, we have 

""" 



By the initial conditions of the problem, that is, i = io 1 for x = / = 0, 
there results 



or A2 = 
and hence the temperature at the braking surface is 

iol. 
By the Bromwich integral Eq. (321), 

The " indicial admittance " then is 



The law of heat generation must be such that i = io fort = and 
i = for / = /o, where /o is the time required to stop the machine. 
Moreover, the power transfer into the heat band is assumed to be 
linear. Consequently, 

. = io(to - Q 

/o 
By use of (333) we have 

* - 2 

or 



^ + f 2 V4 <' - 

CTT JQ ^irC 



The surface temperature eo is a maximum for any particular pressure 

when / = ~; hence, , - 

2 2 . /2.R/0 ,^,, 

Max. e Q = 7*o\/"~7r- (343) 

3 * TT 



294 APPLICATIONS OF OPERATIONAL CALCULUS 

It is now necessary to derive an expression for the total energy to 
be dissipated. This energy evidently is the kinetic energy of the rotor 
plus the energy generated during braking due to leakage. 



K.E. of the rotor = *' 69 ^VV"'"' = 4 ' 48 X 1()8 ft ~ lb ' 

r 

Leakage energy = 200,000 / Ndt = 100,000 

A) 

Since the acceleration is supposed to be uniform, tne energy dissipated 
by the brakes is 

A r , pAifrNo 

pAifr I Ndt = t Ql 

/o 2 

where p = pressure at brake surface in pounds per square inch, 
Ai = total area covered by brake shoes, 
A% = total area of brake plate, 

/ = coefficient of friction, 

r = mean radius in feet, 
NQ = initial speed of rotor in radians per second, 

to = time in seconds required to stop the machine. 

Equating energies, we have 

= 4.48 X 10 8 + 100,000 Noto. 

The table of constants is : 

Ai = 20(9 X 18) = 3240 in. 2 

A 2 = 2w X 171 X 10(2.54) 2 = 69,300cm. 2 

No = 88.3 r.p.m. = 9.24 radians per sec. 

pAifr = 3240 X 0.25 X p = 11,540 p Ib. ft., 

where p is in pounds per square inch. Thus, from the equation of 
energies above, 

_ 8.41 X 10 3 
* ~ p - 17.33* 

Substituting this value of to in (343), we have 

2 . ri.682 X 10 4 1* 
Max. eo = - *o -77- :r rr 

3 LirC(p 17.33) J 



SWITCHING 295 

The constants of this equation are: 



- - 46 (69 - 300 cm - 2 > = 31 - 800 

J\ CITl. v_x 



C = 3.9 1 ( 6 9 ( 300cm.*) = 270,000^^:. 

cm. d L. cm. 

to - 11,540 ft-lb. X 9.24 ^^- S = 106,600 /> 

sec. sec. 

* 144,700 p watts. 

Finally then, 

A/T 76.0 j) 

Max. eo = / - 

Vp - 17.33' 

This is the result required. 

Maximum eo is a minimum if p = 34.7 Ib. per sq. in., and for this 
pressure 

Max. e Q = 634 C. 
and 

J 487 sec. = 8.12 min. 

For a brake pressure of 100 Ib. per sq. in. 

Max. e - 836 C. 
and 

to = 1.69 min. 

129. Switching. Mathematically, those phases of the operational 
calculus so far studied deal with the application of the unit e.m.f. at 
/ = 0. Physically, this is accomplished by closing a switch in that 
mesh of the circuit which contains a battery. Since the operational 
calculus handles such a case so effectively, we are led to inquiie 
whether it will also handle cases where switches are opened or where 
switches not in the battery mesh are closed. 

Consider the circuit shown in Fig. 82. At / = 0, the circuit was 
energized by the insertion of a unit e.m.f. at B. By means of the 
methods developed above, the voltage appearing 
at the terminals A may be calculated as a func- 
tion of time. It is apparent that a generator 
supplying this particular voltage may be connected 
across A without affecting the currents in any part IG ' 

of the circuit. If, at / = /i, a voltage, equal in magnitude but 
opposite in sign, be added in series with this generator, the net effect 




296 APPLICATIONS OF OPERATIONAL CALCULUS 

will be to produce zero voltage across A. This is exactly the result 
of closing a switch at A at time / = t\. Evidently the resulting 
current in any mesh may be found by superposing the current due to 
the reversed generator at A upon that which would have existed had 
A been left open. 

EXAMPLE 1. An inductance L is in series with a resistance R. At 
time / = 0, a steady voltage E is inserted into the circuit. At time 
/ = /i, the resistance is short-circuited. Find the current at any 
instant. 

The impedance operator for the original circuit is Z(p) = R + Lp, 
whence the original current is 



where a = R/L. The voltage across the resistance is 
ei = iiR = E(l - O- 

If time be measured from / = t\ instead of t = 0, this voltage would 
become 



If a generator producing a voltage 62 be placed in the circuit instead 
of R, the current will be unchanged. The effect of short-circuiting R 
is then to introduce a generator giving + 62 volts. The resistance 
of the battery being negligible, the impedance operator for 62 is 
Z(p) = Lp. That is, 

1 



This may be evaluated by means of the superposition theorem, i.e., 

A(t) = j-i = t/L, 
Lp 

e(f) =62= E(l - e~ a(t+t *). 
Then 

i 2 = e(G)A(t) + f A(t- \)e'(\)d\ 
JQ 

f? (1 - -*) + r^. 

L JQ L 



SWITCHING 297 

The zero of time must now be returned to t = 0. This is accomplished 
by writing t t\ for /. Then 



This is the increment of current due to short-circuiting the resistance. 
The total current is therefore 



when t > t\. 

The current increases linearly with time, as would be expected. 

The problem of opening a switch at some point in a circuit is not 
fundamentally different from that of closing a switch. In this case, 
however, a current generator rather than a voltage generator must be 
inserted. 

EXAMPLE 2. Suppose that in the circuit of the previous example 
the resistance is initially in the circuit but short-circuited, and at t = t\ 
this short circuit is removed. We desire the current at any instant. 
The initial current is 

T? 

to = 1 = Et/L, 
Lp 

or referred to the time /i, 

i'o = E(t + ti)/L. 

The current io flows through the switch until t = /i, at which time the 
switch is opened. This is equivalent to inserting a generator producing 
a current i'o at / = 0. This generator acts on a circuit consisting of 
R and L in parallel. The impedance operator for this circuit is 



R + Lp a 
The voltage across the parallel circuit is 

e = i'oz 
whence the current through the resistance is 

in = E/R = i'o 



This expression may be evaluated by the superposition theorem. 
Although this theorem was derived for current in terms of indicial 



298 APPLICATIONS OF OPERATIONAL CALCULUS 

admittance, it can also be applied to give the current in terms of the 
impedance function as in this case. Here 

A(t)=Z(p) 1 = -1 = Re~ 
a 



e(X) = i'oOO = E(\ + h)/L 
whence 






/* 
Eae 
_ 



(ah - l)c-*]. 
Shifting the time axis back to / = 0, 

i' R = [1 + (ah - l)- atf -*>J. 

This is the increment of current through R due to opening the switch. 
It is to be added to the current originally present in R; since R was 
originally short-circuited, the latter current is zero, and I'R is the 
actual current in the circuit after t = t\. 

EXERCISES AND PROBLEMS 
Prove the two shifting formulas: 



1. 

2. 


1 ,,-frf 4 -*t * 


1 
</ 


Z(p) Z(p - b) 

1 1 ,-*' ^ ! 


Z(p) p - 6 Z(p - 


t V * 



3. In transoceanic cables, it is customary to " load " the line by either inserting 
inductance coils at frequent intervals along the length or by wrapping the wire with 
a magnetic tape. In this case, the leakage is small, but L, R, and C must all be con- 
sidered. The entering current then is 



Find io as a function of the time. 

4. In the circuit of the Fig. 82a, the switch S is closed at time t = and Si H? 
closed at time /o. Find the current at time t subsequent to J . 



SWITCHING 



299 



R 

-AAMr 



FIG. 82a. 

5. Solve the example of 128 subject to the assumptions: 

(a) That the thickness of the brake plate has an effect on the surface tem- 
perature. 

(b) The loss of heat from the sides of the brake plate is negligible. 
Consider the problem to be equivalent to an open-circuited transmission line 

whose length is 6. (See example 2, 125.) 

6. From the results of problem 5 plot a curve of temperature in degrees Centi- 
grade of the brake plate surface against time, after applying the brakes, up to the 
time when the machine stops. Use the value of the thickness b and the hydraulic 
cylinder pressure given below. 



b inches 


Pressure on Brake 
Plate Surface, Ib. per 
sq. in. 


b inches 


Pressure on Brake 
Plate Surface, Ib. per 
sq. in. 


2.00 


55 


2.00 


200 


2.00 


60 


0.50 


100 


2.00 


70 


0.75 


100 


2.00 


80 


1.00 


100 


2,00 


90 


1.25 


100 


2.00 


100 


1.50 


100 


2.00 


110 


1.75 


100 








* 


2.00 


120 


2.25 


100 


2.00 


130 


2.50 


100 


2.00 


140 


3.00 


100 


2.00 


150 


3.50 


100 


2.00 


160 


4.00 


100 


2.00 


170 


5.00 


100 


2.00 


180 


6.00 


100 


2.00 


190 







The operational calculus is not applicable in the analysis of non- 
linear circuits. For a new theory of non-linear circuits containing 
variable inductance, variable capacitance, and variable resistance see 
Ref. 6 at the end of this volume. 



300 REFERENCES TO ADDITIONAL AND RELATED MATERIAL 



REFERENCES TO ADDITIONAL AND RELATED MATERIAL OF THE TEXT 

I. Differential Equations 



TOPIC 

1. Exact Differential Equations and In- 

tegrating Factors. 

2. Homogeneous Linear Equation. 



3. Transformation of Second-order 

Equation into Integrable Form by 
Change of Dependent and Inde- 
pendent Variable. 

4. Integration in Series of Differential 

Equations. 

5. Graphical and Numerical Integration 

of Differential Equations. 

6. Integration of Analytic Non-linear 

Differential Equations. 

7. Set-up of Differential Equations of 

Motion of Complicated Mechan- 
ical and Electrical Systems by 
Lagrange's Equations and Ten- 
sors. 

8. Derivation of the Differential Equa- 

tions of Vibrating Systems. 

9. Linear Transmission Line Circuits. 



10. Analogies between Mechanical and 
Electrical System. 



PUBLICATION 

Murray, Introductory Course in Differen- 
tial Equations , p. 17. 

Kells, Elementary Differential Equations, 
p. 29. 

Murray, loc. cit., p. 82. 

Kells, loc. cit., p. 117. 

Murray, loc. cit., p. 114. 



Murray, loc. cit., p. 101. 
Kells, loc. cit., p. 120. 
See Sec. VI of Chap. II. 

Keller, Mathematics of Modern Engineer- 
ing, Vol. II. 
Keller, loc. cit. 



Kimball, Vibration Prevention in Engi- 
neering (General Electric Co. Series). 

Johnson, Transmission Circuits of Tele- 
phone Communication (Bell Lab. 
Series). 

Nickle, " Oscillographic Solution of 
Electromechanical Systems," Trans. 
A. I.E.E., 54 (1925), p. 847. 



n. Determinants 



11. Proof of Laplace's Expansion. 

12. Factoring Determinants. 

13. Derivative of Determinants. 

14. Symmetric, Skew, and Skew-Sym- 

metric Determinants. 



15. Inverse or Reciprocal Determinants. 



Dickson, Elementary Theory of Equa- 
tions, p. 141. 

Scott and Matthews, Theory of Determi- 
nants, p. 75. 

Scott and Matthews, loc. cit., p. 39. 

Scott and Matthews, loc. cit., p. 88. 

Muir, Theory of Determinants, Vol. I, p. 
395. 

Burnside and Panton, Theory of Equa- 
tions, Vol. II, p. 43. 

Burnside and Panton, loc. cit., p. 41. 



DIMENSIONAL ANALYSIS 



301 



TOPIC 
16. Infinite Determinants. 



17. Systems of Fewer than nLinear Equa- 

tions and Systems of More than n 
Linear Equations in n Unknowns. 

18. Matrices. 



PUBLICATION 

Scott and Matthews, loc. tit., p. 120. 

Whittaker and Watson, A Course of 
Modern Analysis, p. 36. 

Dickson, Elementary Theory of Equa- 
tions, p. 144. 

Dickson, Modern A Igebraic Theories, p. 60. 

Dickson, loc. tit., p. 39. 



III. Fourier Series 



19. Use of Fourier Series in Flywheel 
Design. 



20. Proof of the Convergence of Fourier 

Series to the Function /(#). 

21. General Theorems on Convergence, 

Integration, and Differentiation of 
Fourier Series. 

22. Additional Methods of Fourier Anal- 

ysis. 

23. Treatise on Theory of Fourier Series. 



Doherty and Franklin, " Design of Fly- 
wheels for Reciprocating Machinery 
Connected to Synchronous Generators 
or Motors," Trans. A.S.M.E., 42 
(1920), p. 523. 

Carslaw, Fourier Series and Integrals, 
p. 230. 

Carslaw, loc. tit., Chap. VIII. 



Carse and Shearer, A Course in Fourier 1 s 

Analysis and Periodogram Analysis 

(Edinburgh Math. Tracts). 
Hobson, The Theory of Functions of a 

Real Variable and the Theory of 

Fourier's Series, p. 477. 



IV. Solution of Higher-degree and Transcendental Equations 



24. Algebraic Formulas for the Solution 
of Cubic and Quartic Equations. 



25. Graeffe's Method. 



Burnside and Panton, loc. cit., Vol. I, 

p. 115. 
Dickson, Elementary Theory of Equations, 

p. 31. 
Whittaker and Robinson, Calculus of 

Observations. 
Scarborough, Numerical Mathematical 

Analysis, p. 198. 



26. General Reference on Dimensional 

Analysis. 

27. Original Proof of Formulas for 

Change of Units. 

28. Buckingham's Proof of the Theorem. 

29. Tolman's Principle of Similitude. 

30. Heat Transfer in Pipes. 



V. Dimensional Analysis 

Bridgman, Dimensional Analysis. 



Buchholz, Ann. Phys. t 51 (1916), p. 678. 

Buckingham, Phys.Rev., 4 (1914), p. 345. 
Tolman, Phys. Rev., 3 (1914), p. 244. 
Nusselt, Zeitschrift der Verein Deutscher 
Ingenieur, 53 (1909), pp. 1750-1808. 



302 REFERENCES TO ADDITIONAL AND RELATED MATERIAL 



VI. Numerical Method of Solving Differential Equations 



TOPIC 

31. Numerical Integration of Systems of 
Differential Equations. 



32. Differential Analyzer. A New Ma- 
chine for Solving Differential 
Equations. 



PUBLICATION 

Moulton, New Methods in Exterior Bal- 
listics, p. 60. 

Scarborough, Numerical Mathematical 
Analysis, p. 218. 

Adams, Smithsonian Mathematical For- 
mulae and Tables of Elliptic Functions, 
Chapter X. 

Whittaker and Robinson, The Cakulus 
of Observations. 

Bush, Journal of the Franklin Institute, 
(1931), p. 447. 



VII. Vector Analysis 



33. Proofs of Green's, Stokes', and Gauss' 
Theorems. 



34. Scalar Potential. 

35. Vector Potential. 



36. Behavior of Vector Potential 
Flux Density at Boundaries. 



and 



37. Derivation of the Partial Differential 
Equations of Mathematical Phys- 
ics. 



38. Dyadics. 



Coffin, Vector Analysis, pp. 112, 124,148. 
Mason and Weaver, The Electromagnetic 

Field, pp. 102, 348, 349. 
Gibbs, Vector Analysis, pp. 184, 187, 197. 
Kellogg, Foundation of Potential Theory 
Stevenson and Park, General Electric Re. 

view, 31 (1928), pp. 89, 153. 
Mason and Weaver, The Electromagnetic 

Field, p. 191. 
Hague, Electromagnetic Problems in 

Electrical Engineering, p. 77. 
Stevenson, General Electric Review, 29 

(1926), p. 797. 
Mason and Weaver, The Electromagnetic 

Field, pp. 197 and 212. 
Hague, Electromagnetic Problems in Elec- 
trical Engineering, Chap. IV. 
Webster and Szego, Partielle Differential- 

gleichungen der mathematischen Physik, 

Chap. I. 
Weatherburn, Advanced Vector Analysis , 

Vol. 2. 

Gibbs, Vector Analysis, Chap. V. 
Weatherburn, Advanced factor Analysis, 

Chaps. V, VI, VII, and VIII. 



VIII. Operational Calculus 

39. Bromwich's Original Proof of Heavi- Bromwich, London Math. Soc., 15 (1916), 

side's Results. p. 401. 
(Normal Coordinates in Dynamical 
Systems.) 

40. K. W. Wagner's Proof of Heaviside's Wagner, Archiv ftir Elek. t 4 (1916), p. 

Results. 159. 



OPERATIONAL CALCULUS 



303 



TOPIC 

41. Proofs of Operational Calculus The- 

ory by Means of Integral Equa- 
tions. 

42. Proof of the Equivalence of the 

Methods of Bromwich and Carson. 

43. Bromwich's Treatment of Fractional 

Exponents. 

44. Operational Methods in the Solution 

of Partial Differential Equations. 

45. Impulsive Networks. 

46. Extensive Treatment of Transmis- 

sion Line Transients. 

47. Linear Heat Flow; Switching Opera- 

tions; and Bibliography on Opera- 
tional Methods. 



PUBLICATION 

Carson, Electric Circuit Theory and Op- 
erational Calculus. 

March, Bull. Am. Math. Soc.,33 (1927), 
p.311. 

Bromwich, Proc. Camb. Phil. Soc., 20 
(1921), p. 411. 

Jeffreys, Operational Methods in Mathe- 
matical Physics. 

Bush, Operational Circuit Analysis. 

Berg, Heaviside's Operational Cakulus. 

Berg, Heaviside's Operational Cakulus. 



ANSWERS 
Pages 21-22 

2. le" 4- g sin e = 0. 

4. ms" 4- 62.4?rr 2 5 = 0, where $ is the displacement below equilibrium position 
at time /. 

MIS"I k(s f 2 s'i) 4- kiSi 2(^2 s\) = 0. 
6. a(mi 4- m<i)0"i 4- bm 2 fr 2 + (\ 4- m^gOi = 0, 

a0"i 4- be" 2 4- gO* = 0, where g is the acceleration of gravity. 

Page 34 

2. y = Ci*-' + e~' /2 (C 2 sin Vf * 4- C 3 cos V| 0. 

3. y = Cie a < 4- C 2 te" < . 

4. y = e at (C\ sin &/ 4~ ^"2 cos ^/). 

5. y = Cie 2t 4- C 2 e~ 2/ ^g sin 5^. 

6. y == Cie e 4- C 2 e 2 ' 4- %e*<. 

7. y - Cic*' 4- C 2 e 2 ' 4- ^ e*', (* ^ 2). 
^ l ^ 2 ^ k -2 ' ^ ' 

8. y = C\ sin Jfe/ 4~ ^2 cos kt -\- /(Ca sin kt -\- C* cos Jfe/) : / 2 sin Jfe/. 

/2*\H 

9. ^ = 0o cos ^. 

\mr 2 / 

10. j = - / cos ( ' " r } t, Period = ( -^) , Amplitude = /. 

\ m J \31.2rV 

/AH L 

11. ^ = <#>( - I sin ^/-/ 4- 

W \ ' 



Pages 42-44 
1. y - (^ 



/ 4 

2. y = Ci sin / 4- C 2 cos t 4- - 2< e*'. 

ID D cos 2w/ 

3. * = Ci sin at 4- C7 2 cos at 4- ~ r cos oi/ 4 ; ; 4 . 

(a 2 w 2 ) 2(a 2 4w 2 ) a* 

4. y = (Cie* 4- C 2 e-') cos / 4- (C^ + C 4 e~0 sin / 4- A- (sin 3/ 4- cos 3/). 

305 



306 



ANSWERS 

n - 3 



5. y = e-*'(Ci sin # + C 2 cos / 

=, ft - (ICL - 

z 

_ 
n 



sin / 



cos n*), 



n - i 



7, <* = 4 / = ~ 
Iv CL a n 



_ 
(d - n 2 ) 2 + n 2 c 2 ' n (rf - w 2 

8. y - /'(C/^ + C 2 r V 3) + if*. 

9. (a) tan y = C(l - ^) 3 . 

(ft) y - 1 - (Vl-* - C) a . 
(c) cos y = C cos #. 

10. (a) y = (Ci 



- 2 + ^ 
(c) y = Ci^" 8111 - 2 + 2 sin $. 

11. (a) y ~ (Ci + 



+ C 2 r 8 + f^ 2 log / - f 



Pages 52-54 



n,(liiif\ I /vz 

where a = 

acteristic equation 



and mi, (i = 1, 2, 3, 4) are the roots of the char- 



k d p 



~(k d p 



0. 



2. 0i = 

a 2 = 

where wi = 



+ Cja^ 2 ' + 
B + T, w 2 



4- 



4- w 2 )(q 4- 



3. 21 = 5o + 



, 32 - B'o + 



< - i 



ANSWERS 307 

i "I" -^12 -, 



0, m\ t m* are the roots of the characteristic equation and, due to the initial 
conditions, 5'o, 1, 5 2 , satisfy the linear equations 

2 2 2 

^V^i = - CiE, ^TmiBi = 0, 
< - i < - i 

4. The general solution of the differential equations is: 

3i = <f* li (A\ sin wi/ + -4 2 cos on/) -f ^Wi sin w 2 / + 4 4 cos o> 2 /) -f gi, 
g 2 f lt (A'i sin coi/ + ^' 2 cos i/) + ^'(^'i sin co 2 / + 4 '4 cos co 2 /) 
where A(jco) is the characteristic determinant with p replaced by jw, and 



= arc cot 



- 2C 2 C 12 (C 2 + C 12 )(L 2 

+( 
real part of characteristic determinant 



' coefficient of imaginary part of characteristic determinant' 

' arc tan - - 



i db jwi, 2 rb jw 2 are the roots of the characteristic equation and the A'i 
are related to A 4 by means of the differential equation for the second branch of the 
circuit. (For methods of obtaining the above roots see 41, 46.) 




7. A:" + - - = 0, where x is the vertical displacement of the upper end of the 
ki z W 

spring and k\ = radius of gyration about B. 



0, 



_ , 



where 2/ is the length of the beam, k is the radius of gyration about the center of 
gravity of the beam, and X is the spring constant of one spring. 



9. e" 



308 



ANSWERS 



10. 9" 4- - (/sin 6 - cos 6) = 0, where/ is the coefficient of friction. 

11. 0" 4- 27.60 = 0, = 0.5833 cos S.25/, period = 1.196 seconds. 

12. = 32.2/ 13.33 = velocity of car at any time after cable breaks until 
car strikes air-cushion. 

dx 



dt 



v = [632.6* - 6218(100 - x)~ A + 40,663]^ = velocity of car after 



striking air-cushion, where x is distance measured downward from the point where 
air-cushion begins to act. 



Page 72 

1. (a) x =-8, y =-7, z = 26. 
(c) No solution. 

1 1 -f 2w' 32 4- 2w' 



21 



21 



(f) 



2. 



+4-+^- 



3 - 



where 



221 



) sin (k)t * 0i)f 

I sin (a>/ <^> 2 ), 
2n( jw)2 22 ( jco) 2 2 i 2 (Jw) f 



(6) * = 1, y = 2, z = 3, 
(J) x = 2 = 0, ^ = 2w'. 
213 30^' 



2 = 



Pages 128-130 

1. (a) 1.357, 1.692, -3.049. 
(6) 1.576, 0.491, -2.067. 
(c) 1.247, -1.802, -0.445. 



21 



^ 

_-L 
i 



4- 



0. 



0. 



= 0. 



5. 



ANSWERS 309 

(J) Six roots are: dbl.875, 4.694, 7.855. 
(e) -1.934. 
(/) 0.607. 

2. (a) -1, 3, 0.2361, -4.236. 

(b) 1, 3, 5, 7. 

(c) 10, -3.40, 3, 2.1. 

(d) -3.450, 1.450, 0.5 1.32*. 

3. -4.861, 4.000, 4.000, -3.254, -0.8851. 

4.0.396, 1.349, -3.745, 1.461 dhtl.641, -0.4607 f 1.285. 

5. -0.3002, -0.1602, -0.09888, -0.0579, -0.002084. 

6. Yes. Moreover, the roots are -4.35 20.2*, -8.39 6.36*. 

Pages 205-206 

1. 24.5. 

2. -27.6, 40i - 18 j - 16fc. 
34.7, -60i + 3.9j - 6.9k. 



Paga 234 

3. A = - j {(a - )(a - y) log [(a - *) + (a - y)'] 
+ (a + x)(a - y) log [(a + x)* + (a - ;y)*] 
-f (a - *)(a + y) log [(a - *)* + (a + y) 2 ] 
4- (a + *)(a + y) log ((a + ^) a + (a + y) 2 ] 

r a . y a -f" y I 

-f- (a x)* \ arctan -- h arctan - I 
L a - x a x J 

+ (a + *) 2 arctan ^^ + arctan ?LJ! 
[ a + * a + x J 



/ x. F 

\ a ~" y)\ 
L 



, 
arctan -- h arctan 

a - y a - 



+ (a + y) 2 arctan -- h arctan I , 
L a + y a-fyJJ 

a side of the square is 2a and P(x, y) is any point exterior to the square 



Page 239 
1. Y - |7 + 6(J X *). 

Page 260 



310 ANSWERS 

Page 276 

12. ii 10.0 - 9.55*- n - 88 ' - 0.45e- 242 ', 
ta - 3.03(~ 2 - 42 ' -e- 11 - 881 ). 

Pages 298-299 

(fl//4) (fl//4) 
(I!) 2 (2!) 2 

where J is the Bessel function of order zero and i = V 1. 

g bt TO 

4. f = -- sin w/ -f | e~ (6 " &l)r ~ &1< sin o>/ sin w(/ * ) 

R f'-** 
+ 7T 2 / -*-<- & i)'o - - i)X gil| w(/ ... / _ X ) [a, cos w(X + / ) 

JL co ' 


6 sin w(X 



-_, l __ 



INDEX 



Numbers refer to pages 



Acceleration, 13 

Addition of vectors, 191 

Algebraic equations, solution of, 95, 105 

Ampere's law, 213, 217 

Analytic function, 248 

Area, as a line integral, 247 

as a vector, 195 
Armature voltage build-up, 167 

Ballistic coefficient, 179 
Ballistic equations, 179 
Boundary conditions, ordinary differ- 
ential equations, 11 
partial differential equations, 221 
Branch points, 269 
Bromwich line integral, 266 
Buckingham TT theorem, 145 

Capacity, 48 

Cauchy-Riemann equations, 248 

Cauchy's theorems, 249, 250 

Change of units, 139 

Circuital theorem, 213, 217 

Closing a switch, 296 

Coefficient, of capacity, 48 
of induction, 48 
of mutual induction, 51 
of thermal conductivity, 210 

Cofactor, 68 

Complementary function, 26 

Complete physical equation, 138 

Complex variable, 243 

Components of a vector, 191 

Cramer's rule, 63 

Criterion for stability, 129 

Curl, 203 

Curvilinear integral, 244 

Damping torque, 17 
Deformable body, 235 
Degree of differential equation, 10 
De Moivre's theorem, 242 



Derivation, of ordinary differential 
equations, 14, 49 

of partial differential equations, 206 
Derivative, directional, 199 

of vectors, 196 
Derived units, 132 
Determinants, expansion of, 57 

Laplace's development of, 58 

minors of, 57 

multiplication of, 62 

properties of, 60 
Difference tables, 177 
Differential equations, degree of, 10 

derivation of, 14, 49 

general solution of, 10 

homogeneous, 22 

Laplace's, 208, 214 

linear, 7 

of electric circuits, 49 

of harmonic motion, 32 

of mechanical oscillations, 14 

partial, 206 

Dimensional analysis, 130 
Dimensional constants, 143, 153 
Dimensional formulas, 132 
Dimensional homogeneity, 138 
Dimensions, 131 
Displacement current, 219 
Divergence, 200 
Dominant equation, 110 
Dyadics, 235 

nonian form, 237 

product with vector, 236 

rotational, 238 

use in electrical machine theory, 239 

Electric circuits, 49, 67 
Electric displacement, 216 
Electric field strength, 215 
Electric moment, 215 
Electromagnetic equations, 219 
Electrostatic potential, 215 



311 



312 



INDEX 



Electrostatic field, 216 
Encke roots, 99 
Engineering functions, 73 
Epoch angle, 32 
Equation of continuity, 202 
Equations, algebraic, 95, 105 

differential, 7 

dimensional, 145 

partial differential, 206 

vector, 206 

Equilibrium configuration, 15 
Equipotential surface, Iv9 
Essential singularity, 255 
Euler's fluid equation, 220 
Expansion theorem, 267 
Expansions, in Fourier series, 77, 79, 
81,82 

Heaviside's, 267 
Extrapolation formula, 178 



Faraday's law, 217 
Field equations, 219 
Field strength, 214, 215 
Fields, electrostatic, 215 

fluid, 220 

magnetic, 214 

of zero curl, 211 
Flow in pipes, 136 
Fluid motion, 220 
Flux, 195 

Flywheel design, 75 
Force, as a vector, 194 

between moving charges, 225 
Forced vibrations, 17, 20 
Fourier series, analysis, 82 

coefficients, 77, 79, 81, 82 

expansion, 77, 79, 81, 82 

differentiation, 91 

integration, 91 

solution of equations with, 93 
Fractional powers of p, 268 
Frequency, 29, 32 

Functions of a complex variable, 250 
Fundamental dimensions, 132 



Gamma function, 272 
Gauss's theorem, 207 
Gradient, 198 



Graeffe's root-squaring method, 98, 105, 

125 
for complex roots, 103, 111, 114, 117, 

127 

for equal roots, 109, 119 
for real roots, 99, 106, 126 
general theory, 105 
summary of rules, 125 
Graphical solution of differential equa- 
tions, method of curvature, 173 
method of isoclines, 165, 170 
Gravitational constant, 153 
Green's formula, 247 
Green's theorem, 207 



Harmonic motion, 32 

Heat-current density, 210 

Heat equation, without sources, 209 

with sources, 210 
Heaviside's calculus, 240 
Heaviside's expansion theorem, 267 
Heaviside's extended problem, 272 
Heaviside's rules, 267, 268, 272 
Heaviside's shifting, 277 
Heaviside's unextended problem, 262 
Heaviside's unit function, 261, 276 
Higher degree equations, 94 
Homogeneous differential equations, 22 
Homogeneous strain, 235 



Ideal cable, 285 
Impedance as a dyadic, 239 
Indicial admittance, 262 
Infinite distortionless line, 282 
Infinity, behavior at, of 
vector potential, 229 
Initial conditions, 9 
Integral, line, 193 

Bromwich's, 266 

potential, 212 

vector potential, 226, 228 
Integral transformations, 2C7 
Integration, numerical, 163 

along a curve, 193 

of Fourier series, 91 

over a surface, 194 

of function of a complex variable, 
252 



INDEX 



313 



Intensity, electric, 215 

magnetic, 214 
Isoclines, 165, 170 
Isothermal, 209 

Kinetic reaction, 12 
Kirchhoff's laws, 48 

Laplace's equation, 208, 214 
Laurent's expansion, 255 
Law, Ampere's, 213, 217 

Gauss's, 213 

Faraday's, 217 

Newton's, 11 
Laws of vectors, 207 
Linear circuits, 49 
Linear differential equations, 7 
Line integrals, 193 

Magnetic field, 226 

Magnetic potential, 227 

Magnetostatics, 214 

Matrix, 152, 154 

Maxwell's equations, 219 

Maxwell's generalization, 218 

Mechanical oscillations, 14 

Minor, 57 

Models, 155, 157, 162 

Modulus of complex number, 243 

Motion, differential equations of, 14, 49 

fluid, 220 

Newton's laws of, 11 

of charge, 225 

simple harmonic, 32 

Newtonian potential, 211, 212, 214 
Newton's interpolation formula, 178 
Newton's laws, 11 
Newton's method of solution, 95 
Non-homogeneous differential equations, 

34,38 

Nonian form, 237 
Non-linear equations, 167, 170, 187; 

circuits, 298 
Normal equations, 170 
rule for writing, 179 
Normal flux, 195 

Number of fundamental units, 132 
Numerical integration of differential 

equations, 163 



Numerical integration of differential 

equations, general method, 176 
simple methods, 165 

Open circuit ideal cable, 285 
Operational calculus, 240 

formulas, 266, 276 

shifting, 277 

Operator, differential, 34, 204 
Order of differential equations, 10 
Ordinary differential equations, 7 

Parallelogram law of addition, 191 
Partial differential equations, 221 

of electromagnetic field, 219 

of fluid flow, 220 

of heat conduction, 209, 210 

of magnetic vector potential, 226. 

228 

Phase angle, 32 
Physical equation, 138 
Pi theorem, 145 
Poles, 254 
Postfactor, 237 
Potential, vector, 211 
electrostatic, 215 
gravitational, 214 
of infinite conductors, 228 
Precision of Graeffe's method, 128 
Prefactor, 237 
Primary quantities, 132 
Projectile equations, 179 
Propeller thrust, 155 

Quadrature formula, 169 
Quadrature, mechanical, 185 
Quotient of complex numbers, 243 

Radius of curvature method, 173 
Rank of matrix, 153 
Refrigerator leakage, 288 
Region, 244 
Residue theorem, 256 
Resistance on airplane wing, 146 
Resonance, 29 
Roots of equations, 94 

Graeffe's method, 98, 105 

Newton's method, 95 
Rotor oscillations, 75, 93, 134